1 | \section{Conversion from range to kinetic energy} |
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2 | |
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3 | \subsection{Charged particles} |
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4 | The algorithm which converts the stopping range of a charged particle to the |
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5 | corresponding kinetic energy is essentially the same for all charged |
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6 | particle types: given the stopping range of a particle, a vector holding the |
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7 | corresponding kinetic energy for every material is constructed. Only the |
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8 | energy loss formulae are different, depending on whether the particle is an |
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9 | electron, a positron, or a heavy charged particle (muon, pion, proton, etc.). |
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10 | \noindent |
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11 | For protons and anti-protons the above procedure is followed, but for other |
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12 | charged hadrons, the cut values in kinetic energy are computed using the |
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13 | proton and anti-proton energy loss and range tables. |
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14 | |
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15 | \subsubsection{General scheme} |
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16 | \begin{enumerate} |
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17 | \item An energy loss table is created and filled for all the elements |
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18 | in the element table. |
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19 | \item For every material in the material table the following steps |
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20 | are performed: |
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21 | \begin{enumerate} |
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22 | \item a range vector is constructed using the energy loss table |
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23 | and specific formulae for the low energy part of the |
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24 | calculations, |
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25 | \item the conversion from stopping range to kinetic energy is |
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26 | performed and the corresponding element of the |
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27 | {\tt KineticEnergyCuts} vector is set, |
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28 | \item the range vector is deleted. |
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29 | \end{enumerate} |
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30 | \item The energy loss table is deleted at the end of the process. |
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31 | \end{enumerate} |
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32 | |
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33 | \subsubsection{Energy loss formula for heavy charged particles} |
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34 | The energy loss of the particle is calculated from a simplified Bethe-Bloch |
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35 | formula if the kinetic energy of the particle is above the value |
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36 | \[ |
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37 | T_{lim}=2 \,MeV \times \left( \frac {\mbox{particle mass}} |
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38 | {\mbox{proton mass}} \right ). |
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39 | \] |
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40 | The word ``simplified'' means that the low energy shell correction term and |
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41 | the high energy Sternheimer density correction term have been omitted. |
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42 | Below the energy value $T_{lim}$ a simple parameterized energy loss formula |
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43 | is used to compute the loss, which reproduces the energy loss values |
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44 | of the stopping power tables fairly well. The main reason for using a |
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45 | parameterized formula for low energy is that the Bethe-Bloch formula breaks |
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46 | down at low energy. The formula has the following form : |
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47 | \[ |
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48 | \frac{dE}{dx} = \left \{ \begin{array}{ll} |
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49 | a*\sqrt{\frac{T}{M}}+b*\frac{T}{M} & \mbox{for $T \in [0, \, T_0]$} |
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50 | \\ \\ |
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51 | c*\sqrt{\frac{T}{M}} & \mbox{for $T \in [T_0, \, T_{lim}]$} |
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52 | \end{array} |
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53 | \right. |
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54 | \] |
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55 | \begin{tabbing} |
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56 | whereb:bbb\= \kill |
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57 | where : \> M = particle mass \\ |
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58 | \> T = kinetic energy \\ |
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59 | \> $ T_0=0.1 \, MeV \times Z^{1/3} \times \mbox{ M/(proton mass)} $\\ |
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60 | \> Z = atomic number. |
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61 | \end{tabbing} |
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62 | The paramaters $a, b$ and $c$ have been chosen in such a way that $dE/dx$ is a |
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63 | continuous function of $T$ at $T=T_{lim}$ and $T=T_0$, and $dE/dx$ reaches its |
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64 | maximum at the correct $T$ value. |
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65 | |
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66 | \subsubsection{Energy loss of electrons and positrons} |
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67 | The Berger-Seltzer energy loss formula has been used for $T > 10$ keV to |
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68 | compute the energy loss due to ionization. This formula plays the role of |
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69 | the Bethe-Bloch equation for electrons (see e.g. the GEANT3 manual). Below |
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70 | 10 keV the simple $c$/($T$/mass of electron) parameterization has been used, |
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71 | where $c$ can be determined from the requirement of continuity at $T = 10$ keV. |
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72 | For electrons the radiation loss is important even at relatively low (few MeV) |
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73 | energies, so a second term has been added to the energy loss formula which |
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74 | accounts for radiation losses (losses due to bremsstrahlung). This second |
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75 | term is an empirical, parameterized formula. |
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76 | For positrons a different formula is used to calculate the ionization loss, |
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77 | while the term accounting for the radiation losses is the same as that |
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78 | for electrons. |
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79 | |
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80 | \subsubsection{Range calculation} |
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81 | The stopping range is defined as |
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82 | \[ R(T)= \int_0^T \frac{1}{(dE/dx)} \, dE \] . |
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83 | The integration has been done analytically for the low energy part and |
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84 | numerically above an energy limit. |
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85 | |
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86 | \subsection{Photons} |
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87 | Starting from a particle cut given in absorption lengths, the method |
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88 | constructs a vector holding the cut values in kinetic energy for every |
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89 | material. The main steps of the algorithm are the following : |
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90 | \subsubsection{General scheme} |
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91 | \begin{enumerate} |
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92 | \item A cross section table is created and filled for all the elements |
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93 | in the element table. |
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94 | \item For every material in the material table the following steps are |
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95 | performed: |
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96 | \begin{enumerate} |
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97 | \item an absorption length vector is constructed using the |
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98 | cross section table |
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99 | \item the conversion from absorption length to kinetic energy is |
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100 | performed and the corresponding element of the |
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101 | {\tt KineticEnergyCuts} vector is set |
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102 | (It contains the particle cut value in kinetic energy |
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103 | for the actual material.), |
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104 | \item the absorption length vector is deleted. |
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105 | \end{enumerate} |
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106 | \item The cross section table is deleted at the end of the process. |
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107 | \end{enumerate} |
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108 | \subsubsection{Cross section formula for elements} |
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109 | An approximate empirical formula is used to compute the {\em absorption |
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110 | cross section} of a photon in an element. Here, the {\em absorption cross |
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111 | section} means the sum of the cross sections of the gamma conversion, Compton |
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112 | scattering and photoelectric effect. These processes are the ``destructive'' |
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113 | processes for photons: they destroy the photon or decrease its energy. |
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114 | (The coherent or Rayleigh scattering changes the direction of the gamma |
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115 | only; its cross section is not included in the {\em absorption cross section}.) |
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116 | |
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117 | \subsubsection{Absorption length vector} |
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118 | The {\tt AbsorptionLength} vector is calculated for every material as : |
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119 | \begin{center} |
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120 | {\tt AbsorptionLength} = 5/(macroscopic absorption cross section) . |
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121 | \end{center} |
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122 | The factor 5 comes from the requirement that the probability of having |
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123 | no 'destructive' interaction should be small, hence |
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124 | \begin{eqnarray*} |
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125 | \exp(-\mbox{{\tt AbsorptionLength * MacroscopicCrossSection}}) &=&\exp(-5) \\ |
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126 | &=& 6.7 \times 10^{-3} |
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127 | \end{eqnarray*} |
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128 | |
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129 | \subsubsection{Meaningful cuts in absorption length} |
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130 | The photon cross section for a material has a minimum at a certain kinetic |
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131 | energy $T_{min}$. The {\tt AbsorptionLength} has a maximum at $T=T_{min}$, |
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132 | the value of the maximal {\tt AbsorptionLength} is the biggest "meaningful" |
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133 | cut in absorption length. If the cut given by the user is bigger than this |
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134 | maximum, a warning is printed and the cut in kinetic energy is set to the |
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135 | maximum gamma energy (i.e. all the photons will be killed in the material). |
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136 | |
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137 | \subsection{Status of this document} |
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138 | \ 9.10.98 created by L. Urb\'an. \\ |
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139 | 27.07.01 minor revision M.Maire \\ |
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140 | 17.08.04 moved to common to all charged particles (mma) \\ |
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141 | 04.12.04 minor re-wording by D.H. Wright \\ |
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142 | |
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