1 | \section{The Interaction Length or Mean Free Path} \label{mfp} |
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2 | \begin{itemize} |
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3 | \item[1)] |
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4 | In a simple material the number of atoms per volume is: |
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5 | \[n = \frac{\mathcal{N}\rho}{A}\] |
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6 | where: |
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7 | \begin{eqnarray*} |
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8 | \mathcal{N} & & \mbox{Avogadro's number} \\ |
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9 | \rho & & \mbox{density of the medium} \\ |
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10 | A & & \mbox{mass of a mole} |
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11 | \end{eqnarray*} |
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12 | \item[2)] |
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13 | In a compound material the number of atoms per volume of the |
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14 | $i^{th}$ element is: |
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15 | \[n_{i} = \frac{\mathcal{N}\rho w_{i}}{A_{i}}\] |
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16 | where: |
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17 | \begin{eqnarray*} |
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18 | \mathcal{N} & & \mbox{Avogadro's number} \\ |
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19 | \rho & & \mbox{density of the medium} \\ |
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20 | w_{i} & & \mbox{proportion by mass of the $i^{th}$ element}\\ |
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21 | A_{i} & & \mbox{mass of a mole of the $i^{th}$ element} |
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22 | \end{eqnarray*} |
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23 | \item[3)] |
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24 | The {\bf mean free path} of a process, $\lambda$, also called the |
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25 | {\bf interaction length}, can be given in terms of the total cross |
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26 | section : |
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27 | $$ |
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28 | \lambda(E) = |
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29 | \left( \sum_i \lbrack n_i \cdot \sigma(Z_i,E) \rbrack \right)^{-1} |
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30 | $$ |
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31 | where $\sigma(Z,E)$ is the total cross section per atom of the |
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32 | process and $\sum_{i}$ runs over all elements composing the material. |
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33 | |
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34 | $\sum\limits_{i}{\lbrack n_{i} \sigma(Z_{i},E)\rbrack}$ is also |
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35 | called the {\it macroscopic cross section}. The mean free path is the |
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36 | inverse of the macroscopic cross section. |
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37 | \end{itemize} |
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38 | |
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39 | \noindent |
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40 | Cross sections per atom and mean free path values are tabulated during |
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41 | initialisation. |
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42 | |
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43 | \section{Determination of the Interaction Point} \label{ip} |
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44 | The mean free path, $\lambda$, of a particle for a given process depends on the |
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45 | medium and cannot be used directly to sample the probability of an interaction |
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46 | in a heterogeneous detector. The number of mean free paths which a particle |
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47 | travels is: |
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48 | |
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49 | \begin{equation} |
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50 | \label{int.c} |
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51 | n_\lambda =\int_{x_1}^{x_2} \frac{dx}{\lambda(x)} , |
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52 | \end{equation} |
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53 | which is independent of the material traversed. If $n_r$ is a random variable |
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54 | denoting the number of mean free paths from a given point to the point of |
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55 | interaction, it can be shown that $n_r$ has the distribution function: |
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56 | \begin{equation} |
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57 | \label{int.d} |
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58 | P( n_r < n_\lambda ) = 1-e^{-n_\lambda} |
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59 | \end{equation} |
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60 | The total number of mean free paths the particle travels before reaching |
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61 | the interaction point, $n_\lambda$, is sampled at the beginning of the |
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62 | trajectory as: |
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63 | \begin{equation} |
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64 | \label{int.e} |
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65 | n_\lambda = -\log \left ( \eta \right ) |
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66 | \end{equation} |
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67 | where $\eta$ is a random number uniformly distributed in the range $(0,1)$. |
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68 | $n_\lambda$ is updated after each step $\Delta x$ according the formula: |
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69 | \begin{equation} |
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70 | \label{int.f} |
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71 | n'_\lambda=n_\lambda -\frac{\Delta x }{\lambda(x)} |
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72 | \end{equation} |
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73 | until the step originating from $s(x) = n_\lambda \cdot \lambda(x)$ is |
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74 | the shortest and this triggers the specific process.\\ |
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75 | |
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76 | \noindent |
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77 | The short description given above is the {\em differential approach} to |
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78 | particle transport, which is used in most simulation |
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79 | codes (\cite{int.egs4},\cite{int.geant3}). |
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80 | In this approach besides the other ({\em discrete}) processes |
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81 | the continuous energy loss imposes a limit on the stepsize too, because the |
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82 | cross sections depend of the energy of the particle. |
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83 | Then it is assumed that the step is small enough so that the particle cross |
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84 | sections remain approximately constant during the step. |
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85 | In principle one must use very small steps in order to insure an accurate |
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86 | simulation, but computing time increases as the stepsize decreases. A good |
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87 | compromise is to limit the stepsize in Geant4 by not allowing the stopping |
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88 | range of the particle to decrease by more than 20 \% during the step. This |
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89 | condition works well for particles with kinetic energies $>$ 0.5 MeV, but for |
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90 | lower energies it can give very short step sizes. To cure this problem a lower |
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91 | limit on the stepsize is also introduced. |
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92 | |
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93 | \section{Updating the Particle Lifetime} |
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94 | |
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95 | The proper and laboratory times of the particle should be updated after each |
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96 | step. |
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97 | In the laboratory system: |
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98 | \begin{equation} |
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99 | \label{int.n} |
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100 | \Delta t_{lab} = \frac{\Delta x}{0.5 (v_0 + v)} |
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101 | \end{equation} |
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102 | where |
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103 | \[ |
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104 | \begin{array}{ll} |
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105 | \Delta x & \mbox{step travelled by the particle} \\ |
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106 | v_0 & |
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107 | \mbox{particle velocity at the beginning of the step} \\ |
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108 | v & |
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109 | \mbox{particle velocity at the end of the step} \\ |
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110 | \end{array} |
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111 | \] |
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112 | This expression is a good approximation if the velocity is not allowed to |
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113 | change too much during the step. |
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114 | |
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115 | |
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116 | \section{Status of this document} |
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117 | 09.10.98 created by L. Urb\'an. \\ |
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118 | 27.07.01 minor revisions by M. Maire \\ |
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119 | 01.12.03 integral method subsection added by V. Ivanchenko \\ |
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120 | 12.08.04 splitted and partly moved in introduction by M. Maire \\ |
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121 | 25.12.06 minor revision by V. Ivanchenko \\ |
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122 | |
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123 | \begin{latexonly} |
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124 | |
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125 | \begin{thebibliography}{99} |
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126 | \bibitem{int.egs4} W.R. Nelson et al. |
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127 | the \textsc{egs4} Code System. SLAC-Report-265, December 1985. |
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128 | |
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129 | \bibitem{int.geant3} |
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130 | G\textsc{eant3} manual, CERN Program Library Long Writeup W5013 (October 1994). |
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131 | |
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132 | \end{thebibliography} |
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133 | |
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134 | \end{latexonly} |
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135 | |
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136 | \begin{htmlonly} |
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137 | |
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138 | \section{Bibliography} |
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139 | |
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140 | \begin{enumerate} |
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141 | \item W.R. Nelson et al. |
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142 | the \textsc{egs4} Code System. SLAC-Report-265, December 1985. |
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143 | |
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144 | \item G\textsc{eant3} manual, CERN Program Library Long Writeup W5013 (October 1994). |
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145 | |
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146 | |
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147 | \end{enumerate} |
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148 | |
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149 | \end{htmlonly} |
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