1 | \section{Hadron and Ion Ionisation} \label{le_had_ion} |
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2 | |
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3 | |
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4 | The class {\tt G4hLowEnergyIonisation} calculates the continuous energy loss |
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5 | due to ionisation and simulates the $\delta$-ray production by charged hadrons |
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6 | or ions. This represents an extension of the Geant4 physics models down to |
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7 | low energy \cite{hlei.prepHadr,hlei.prepIon}. |
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8 | |
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9 | \subsection{Delta-ray production} |
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10 | |
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11 | In Geant4, $\delta$-rays are generated generally only above a threshold |
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12 | energy, $T_c$, the value of which depends on atomic parameters and the cut |
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13 | value, $T_{cut}$, calculated from the unique {\em cut in range} parameter |
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14 | for all charged particles in all materials. The total cross-section |
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15 | for the production of a $\delta$-ray electron of kinetic energy $T > T_c$ |
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16 | by a particle of kinetic energy $E$ is: |
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17 | \begin{equation} |
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18 | \label{hlei.a} |
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19 | \sigma (E,T_{c}) = \int_{T_{c}}^{T_{max}} \frac{d \sigma (E,T)}{dT} dT |
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20 | \hspace{5mm} \mbox{with } T_c = \min(\max(I,T_{cut}),T_{max}) |
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21 | \end{equation} |
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22 | where $I$ is the mean excitation potential of the atom (the formulae of |
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23 | this charter are precise if $T \gg I$), |
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24 | $T_{max}$ is the maximum energy transferable to the free electron |
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25 | \begin{equation} |
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26 | \label{hlei.a1} |
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27 | T_{max} =\frac{2m_e c^2 (\gamma^2 -1)} {1+2\gamma (m_e/M) + (m_e/M)^2} |
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28 | \end{equation} |
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29 | with $m_e$ the electron mass, $M$ the mass of the incident particle, and |
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30 | $\gamma$ is the relativistic factor. |
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31 | For heavy charged particles the differential cross-section per atom |
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32 | can be written as \cite{hlei.pdg,hlei.rossi52}: |
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33 | \begin{eqnarray} |
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34 | \label{hlei.bbb} |
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35 | \mbox{for spin 0} &\frac {d\sigma }{dT} = & K Z \frac {Z^2_h}{\beta^2 T^2} |
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36 | \left[ 1- \beta^2 \frac{T} { T_{max} }\right] |
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37 | \\ \nonumber |
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38 | \mbox{for spin 1/2} &\frac{d \sigma} {dT} = & K Z \frac {Z^2_h} {\beta^2 T^2} |
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39 | \left[1- \beta^ 2 \frac{T}{T_{max} }+ \frac{T^2} {2E^2} |
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40 | \right] |
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41 | \\ \nonumber |
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42 | \mbox{for spin 1} &\frac{d \sigma} {dT}= & K Z \frac {Z^2_h}{\beta^2 T^2} |
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43 | \left[\left(1- \beta^ 2 \frac{T}{T_{max} }\right) |
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44 | \left(1 + \frac{T}{3Q_c} \right) |
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45 | + \frac{T^2} {3E^2}\left(1+\frac{T}{2Q_c}\right) |
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46 | \right] |
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47 | \end{eqnarray} |
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48 | where $Z$ is the atomic number, |
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49 | $Z_{h}$ is the effective charge of |
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50 | the incident particle in units |
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51 | of positron charge, $\beta$ is the relativistic velocity, and |
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52 | $Q_c=(M c^2)^2/m_e c^2$. The |
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53 | factor $K$ is expressed as |
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54 | $K = 2\pi r^2_e m_e c^2$, |
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55 | where $r_e$ is the classical electron |
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56 | radius. |
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57 | The integration of |
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58 | formula (\ref{hlei.a}) gives the total cross-section, |
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59 | which |
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60 | for particles with spin 0 and 1/2 are the following : |
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61 | \begin{eqnarray} |
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62 | \label{hlei.c} |
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63 | \mbox{for spin 0} &\sigma (Z,E,T_{c}) = & |
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64 | K Z \frac{Z^2_h}{\beta^2} \left ( |
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65 | \frac{1-\tau+\beta^2 \tau\ln \tau}{T_{c}} \right ) |
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66 | \\ \nonumber |
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67 | \mbox{for spin 1/2} &\sigma (Z,E,T_{c}) = & |
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68 | K Z \frac{Z^2_h}{\beta^2} \left ( \frac{1-\tau+\beta^2 \tau\ln \tau}{T_{c}} |
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69 | + \frac{T_{max}-T_{c}} {2E^2} \right ) |
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70 | \end{eqnarray} |
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71 | where $\tau = T_c/T_{max}$. |
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72 | |
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73 | \noindent |
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74 | The average energy transfer $\Delta E_{\delta}$ |
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75 | of a particle with spin 0 |
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76 | to $\delta$-electrons with $T > T_c$ |
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77 | can be expressed as: |
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78 | \begin{equation} |
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79 | \Delta E_{\delta} = |
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80 | N_{el}\frac{Z^2_h}{\beta^2} \left (- |
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81 | \ln{\tau} - |
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82 | \beta^2(1-\tau) \right ) |
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83 | \label{hlei.del} |
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84 | \end{equation} |
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85 | where $N_{el}$ is the electron density of the medium. |
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86 | Using (\ref{hlei.bbb}) one finds that |
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87 | the correction to (\ref{hlei.del}) for particles with spin 1/2 |
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88 | is $(T^2_{max}-T^2_c)/4E^2$. |
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89 | This value is very small for low energy and can be neglected. |
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90 | The same conclusion can be drawn for particles with spin 1. |
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91 | |
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92 | \noindent |
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93 | The mean free path of the particle |
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94 | is tabulated during initialisation |
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95 | as a function of the material and of the energy for |
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96 | all the charged hadrons and static ions. Note, that for |
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97 | low energy $T_c = T_{max}$, cross-section is zero and |
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98 | the mean free path is set to infinity, compatible with the |
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99 | machine precision. |
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100 | |
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101 | |
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102 | \subsection{Energy Loss of Fast Hadrons} |
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103 | |
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104 | |
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105 | The energy lost in soft |
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106 | ionising collisions producing $\delta$-rays below ${T_c}$ |
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107 | are included in the continuous energy loss. |
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108 | The mean value of the energy loss |
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109 | is given by the restricted Bethe-Bloch formula \cite{hlei.bethe,hlei.pdg} : |
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110 | \begin{eqnarray} |
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111 | \left.\frac{dE}{dx} \right]_{T<T_c} &=& K N_{el}\frac{Z^2_{h}}{\beta^2}L_0 |
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112 | \\ \nonumber |
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113 | &=& K N_{el}\frac{Z^2_{h}}{\beta^2} \left [ |
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114 | \ln{\frac{2m_e c^2 \beta^2\gamma^2T_{max}}{I^2}} - |
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115 | \beta^2 \left ( 1 +\frac{T_c}{T_{max}} \right ) |
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116 | - \delta - \frac{2C_e}{Z} \right ] |
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117 | \label{hlei.d} |
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118 | \end{eqnarray} |
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119 | where $N_{el}$ is the electron density of the medium, |
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120 | $\delta$ is the density correction term, and $C_e/Z$ is the shell correction |
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121 | term. |
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122 | |
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123 | \noindent |
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124 | The density effect becomes important at high |
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125 | energies because of the long-range polarisation |
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126 | of the medium by a relativistic charged particle. The shell correction term |
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127 | takes into account the fact that, at low energies for light elements, |
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128 | and at all energies for heavy ones, the probability of |
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129 | hadron interaction with inner atomic shells becomes small. |
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130 | The accuracy of the Bethe-Bloch formula with the correction terms mentioned |
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131 | above is estimated as 1~\% for |
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132 | energies between 6~MeV and 6~GeV \cite{hlei.pdg}. |
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133 | Using (\ref{hlei.bbb}) one can find out that |
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134 | the correction to $L_0$ for particles with the spin 1/2 |
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135 | is $T^2_c/4E^2$. This value is very small and can be neglected. |
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136 | |
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137 | \noindent |
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138 | There exists a variety of phenomenological approximations for |
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139 | parameters in the Bethe-Bloch formula. |
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140 | In Geant4 the tabulation of |
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141 | the ionisation potential from Ref.\cite{hlei.ICRU37} |
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142 | is implemented for all the |
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143 | elements. For the density |
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144 | effect the formulation of Sternheimer \cite{hlei.sternheimer} |
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145 | is used: |
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146 | \input{electromagnetic/utils/densityeffect} |
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147 | |
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148 | \noindent |
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149 | The semi-empirical formula due to Barkas, which is applicable to all |
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150 | materials, is used for the shell correction term\cite{hlei.bark62}: |
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151 | \begin{equation} |
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152 | C_e(I, \beta\gamma) = \frac{a(I)}{(\beta\gamma)^2} |
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153 | +\frac{b(I)}{(\beta\gamma)^4} |
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154 | +\frac{c(I)}{(\beta\gamma)^6} |
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155 | \end{equation} |
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156 | The functions a(I), b(I), c(I) can be found in the source code. \\ |
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157 | This formula breaks down at low energies, and it only applies for $\beta |
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158 | \gamma > 0.13$ (e.g. $T > 7.9$ MeV for a proton). |
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159 | For $\beta \gamma \leq 0.13$ the shell correction term is calculated as: |
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160 | $$ |
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161 | \left . C_{e}(I,\beta \gamma) \rule{0mm}{5mm} \right |_{\beta \gamma \leq |
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162 | 0.13} = C_{e}(I,\beta \gamma=0.13)\frac{\ln (T/T_{2l})} |
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163 | {\ln (7.9 \mbox{ MeV}/T_{2l})} |
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164 | $$ |
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165 | hence the correction becomes progressively smaller from $T=7.9$ |
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166 | MeV to $T=T_{2l}=2 \mbox{ MeV}$. |
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167 | |
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168 | \noindent |
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169 | Since $M \gg m_e$, |
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170 | the ionisation loss does not depend on the hadron |
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171 | mass, but on its velocity. |
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172 | Therefore the energy loss of a charged hadron |
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173 | with kinetic energy, $T$, is the same as |
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174 | the energy loss of a proton with the same velocity. The corresponding |
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175 | kinetic energy of the proton $T_p$ is |
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176 | \begin{equation} |
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177 | T_{proton} = \frac{M_{proton}}{M} \ T. |
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178 | \label{hlei.e} |
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179 | \end{equation} |
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180 | |
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181 | \noindent |
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182 | At initialisation stage of Geant4 the $dE/dx$ tables |
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183 | and range tables for all materials |
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184 | are calculated only for protons and antiprotons. |
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185 | During run time the energy loss and the range of any hadron or ion are |
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186 | recalculated using the scaling relation (\ref{hlei.e}). |
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187 | |
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188 | |
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189 | \subsection{Barkas and Bloch effects} |
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190 | |
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191 | |
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192 | The accuracy of |
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193 | the Bethe-Bloch stopping power formula |
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194 | (\ref{hlei.e}) can be improved |
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195 | if the higher order terms are taken into account: |
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196 | \begin{equation} |
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197 | -\frac{dE}{dx} = K \frac{Z^2_{h}}{\beta^2}(L_0 +Z_{h}L_1+Z^2_{h}L_2), |
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198 | \label{hlei.f} |
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199 | \end{equation} |
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200 | where $L_1$ is the Barkas term \cite{hlei.bark56}, |
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201 | describing the difference |
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202 | between ionisation of positively and negatively charged particles, and |
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203 | $L_2$ is the Bloch term. |
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204 | |
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205 | The Barkas effect for kinetic energy of |
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206 | protons or antiprotons greater than $500 keV$ can be described as |
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207 | \cite{hlei.arb72}: |
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208 | \begin{equation} |
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209 | L_1=\frac{F\left ( b / \sqrt{x}\right ) }{\sqrt{Z x^3}}, \,\,\, |
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210 | x=\frac{\beta^2c^2}{Zv_0^2},\,\,\, |
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211 | b=0.8 Z^{\frac 16}\left( 1+6.02Z^{-1.19}\right), |
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212 | \label{hlei.g} |
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213 | \end{equation} |
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214 | where |
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215 | $v_0$ is the Bohr velocity (corresponding to proton energy $T_p=25 keV$), and |
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216 | the function $F$ is tabulated according to \cite{hlei.arb72}. |
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217 | |
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218 | The Bloch term \cite{hlei.bloch} |
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219 | can be expressed in the following way: |
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220 | \begin{equation} |
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221 | Z^2_{h}L_2 = - y^2 \sum^{\inf}_{j=1} \frac{1}{j(j^2 + y^2)},\,\,\, |
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222 | y=\frac{Z_{h}}{137\beta}. |
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223 | \label{hlei.h} |
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224 | \end{equation} |
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225 | Note, that for $y \ll 1$ the simplified expression |
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226 | $Z^2_{h}L_2=-1.202y^2$ can be used. |
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227 | |
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228 | Both the Barkas and Bloch terms break scaling of ionisation losses |
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229 | if the absolute value of particle charge is different from unity, |
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230 | because the particle charge $Z_h$ is not factorised |
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231 | in the formula (\ref{hlei.f}). |
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232 | To take these terms into account correction is made at |
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233 | each step of the simulation for the value of $dE/dx$ |
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234 | re-calculated from the proton or antiproton tables. |
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235 | There is the possibility to switch off the calculation |
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236 | of these terms. |
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237 | |
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238 | \subsection{Energy losses of slow positive hadrons} |
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239 | |
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240 | At low energies the total energy loss is usually described |
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241 | in terms of {\it electronic stopping power} $S_e = - dE/dx$. |
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242 | For charged hadron with velocity $\beta < 0.05$ (corresponding |
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243 | to 1~MeV for protons), formula (\ref{hlei.d}) becomes inaccurate. |
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244 | In this case the velocity of the incident |
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245 | hadron is comparable to the velocity |
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246 | of atomic electrons. At very low energies, when |
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247 | $\beta < 0.01$, the model of a free electron gas \cite{hlei.Lindhard} |
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248 | predicts the stopping power to be proportional to |
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249 | the hadron velocity, |
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250 | but it is not as accurate as the Bethe-Bloch formalism. |
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251 | The intermediate region $0.01 < \beta < 0.05$ is not covered |
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252 | by precise theories. In this energy |
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253 | interval the Bragg peak of ionisation loss occurs. |
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254 | |
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255 | To simulate slow proton energy loss |
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256 | the following |
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257 | parametrisation from the review \cite{hlei.Ziegler771} was implemented: |
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258 | \begin{eqnarray} |
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259 | S_e & = & A_1E^{1/2}, \; \; \; \; \; \; \; \; \hspace{46mm} |
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260 | 1~keV < T_p < 10~keV, \nonumber \\ |
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261 | S_e & = & \frac{S_{low}S_{high}}{S_{low}+S_{high}}, \hspace{46mm} |
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262 | 10~keV < T_p < 1~MeV, \nonumber \\ |
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263 | S_{low} & = & A_2E^{0.45}, \nonumber\\ |
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264 | S_{high} & = & \frac{A_3}{E}\ln{\left(1 + \frac{A_4}{E} + A_5E \right)}, |
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265 | \nonumber \\ |
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266 | S_e & = & \frac{A_6}{\beta^2} \left [\ln{\frac{A_7\beta^2}{1-\beta^2}} |
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267 | -\beta^2 - \sum^{4}_{i=0} A_{i+8}(\ln{E})^i \right ], |
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268 | \; 1~MeV < T_p < 100~MeV, \nonumber \\ |
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269 | \label{hlei.i} |
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270 | \end{eqnarray} |
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271 | where $S_e$ is the stopping power |
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272 | in $[eV/10^{15}atoms/cm^2]$, $E=T_p/M_p [keV/amu]$, $A_i$ are twelve |
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273 | fitting parameters found individually for each atom for |
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274 | atomic numbers from 1 to 92. |
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275 | This parametrisation is used |
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276 | in the interval of proton kinetic energy: |
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277 | \begin{equation} |
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278 | T_1 < T_p < T_2, |
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279 | \label{hlei.j} |
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280 | \end{equation} |
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281 | where $T_1 = 1~keV$ is the minimal kinetic energy of protons |
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282 | in the tables of Ref.\cite{hlei.Ziegler771}, |
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283 | $T_2$ is an arbitrary value |
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284 | between 2~MeV and 100~MeV, since in this range |
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285 | both the parametrisation (\ref{hlei.i}) |
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286 | and the Bethe-Bloch formula (\ref{hlei.e}) |
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287 | have practically the same accuracy and |
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288 | are close to each other. |
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289 | Currently the value $T_2 = 2~MeV$ is chosen. |
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290 | |
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291 | |
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292 | To avoid problems in computation and |
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293 | to provide a continuous $dE/dx$ function, the factor |
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294 | \begin{equation} |
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295 | F = \left (1 + B\frac{T_2}{T_p} \right ) |
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296 | \label{hlei.r} |
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297 | \end{equation} |
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298 | is multiplied by the value of $dE/dx$ for $T_p > T_{2}$. |
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299 | The parameter $B$ is determined for each element of the material |
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300 | in order to |
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301 | provide continuity at $T_p=T_2$. The value of $B$ for all |
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302 | atoms is less than 0.01. For the |
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303 | simulation of the stopping power of very slow protons the model of a |
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304 | free electron gas \cite{hlei.Lindhard} is used: |
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305 | \begin{equation} |
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306 | S_e = A \sqrt{T_p}, \; \; T_p < T_{1}. |
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307 | \label{hlei.k} |
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308 | \end{equation} |
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309 | The parameter $A$ is defined for each atom |
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310 | by requiring the stopping power to be continuous |
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311 | at $T_p=T_{1}$. Currently the value used is $T_1=1~keV$. |
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312 | |
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313 | Note that |
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314 | if the cut kinetic energy is small ($T_c < T_{max}$), then the average |
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315 | energy deposit giving |
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316 | rise to $\delta$-electron production (\ref{hlei.del}) |
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317 | is subtracted from the |
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318 | value of the stopping power $S_e$, which is calculated by formula |
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319 | (\ref{hlei.i}). |
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320 | |
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321 | |
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322 | Alternative parametrisations of proton energy loss |
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323 | are also available within Geant4 (Table \ref{hlei.tab0}). |
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324 | The parameterisation formulae |
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325 | in Ref.\cite{hlei.ICRU49} are the same |
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326 | as in Ref.(\cite{hlei.Ziegler771}) |
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327 | for the kinetic |
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328 | energy of protons $T_p < 1~MeV$, but |
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329 | the values of the parameters are different. |
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330 | The type of parameterisation is optional and |
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331 | can be chosen by the user separately for each particle |
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332 | at the initialisation stage of Geant4. |
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333 | |
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334 | |
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335 | \begin{table*} |
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336 | \caption{The list of parameterisations available.} |
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337 | %\vspace {2pt} |
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338 | \label{hlei.tab0} |
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339 | \begin{center} |
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340 | \begin{tabular}{|l|l|l|} |
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341 | \hline |
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342 | Name & Particle & Source \\ |
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343 | \hline |
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344 | {\bf Ziegler1977p} & proton & J.F.~Ziegler parameterisation |
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345 | \cite{hlei.Ziegler771} \\ |
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346 | {\bf Ziegler1977He} & $He^4$ & J.F.~Ziegler parameterisation |
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347 | \cite{hlei.Ziegler774}\\ |
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348 | {\bf Ziegler1985p} & proton & TRIM'85 parameterisation \cite{hlei.Ziegler85} \\ |
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349 | {\bf ICRU\_R49p} & proton & ICRU parameterisation \cite{hlei.ICRU49} \\ |
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350 | {\bf ICRU\_R49He} & $He^4$ & ICRU parameterisation \cite{hlei.ICRU49} \\ |
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351 | \hline |
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352 | \end{tabular} |
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353 | \end{center} |
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354 | \end{table*} |
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355 | |
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356 | |
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357 | \subsection{Energy loss of alpha particles} |
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358 | |
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359 | The accuracy of the data for the ionisation losses of $\alpha$-particles |
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360 | in all elements \cite{hlei.ICRU49,hlei.Ziegler774} |
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361 | is comparable to the accuracy |
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362 | of the data for proton energy loss \cite{hlei.Ziegler771,hlei.ICRU49}. |
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363 | In the GEANT4 energy loss model for $\alpha$-particles |
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364 | the Bethe-Bloch formula is used for kinetic energy |
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365 | $T > T_2$, where $T_2$ is the arbitrary parameter, currently set to $8~MeV$. |
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366 | For lower energies a parameterisation is performed. |
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367 | In the energy range of the Bragg peak, |
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368 | $1~keV < T < 10~MeV$, the |
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369 | parameterisation is: |
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370 | \begin{eqnarray} |
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371 | S_e & = & \frac{S_{low}S_{high}}{S_{low}+S_{high}}, \nonumber \\ |
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372 | S_{low} & = & A_1T^{A_2}, \nonumber\\ |
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373 | S_{high} & = & \frac{A_3}{T}\ln{\left(1 + \frac{A_4}{T} + A_5T \right)}, |
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374 | \nonumber \\ |
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375 | \label{hlei.l} |
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376 | \end{eqnarray} |
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377 | where $S_e$ is the electronic stopping power |
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378 | in $[eV/10^{15}atoms/cm^2]$, $T$ is the kinetic energy of $\alpha$-particles in |
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379 | $MeV$, |
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380 | $A_i$ are the five fitting |
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381 | parameters fitted individually for each atom for |
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382 | atomic numbers from 1 to 92. |
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383 | |
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384 | For higher energies $T > 10~MeV$, another |
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385 | parametrisation \cite{hlei.Ziegler774} is applied |
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386 | \begin{equation} |
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387 | S_e= exp \left(A_6+A_7E+A_8E^2+A_9E^3 \right ), \; E=ln(1/T). |
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388 | \label{hlei.m} |
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389 | \end{equation} |
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390 | To ensure a continuous $dE/dx$ function from the energy range of the |
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391 | Bethe-Bloch formula to the energy range of the parameterisation, the factor |
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392 | \begin{equation} |
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393 | F = \left (1 + B\frac{T_2}{T} \right ) |
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394 | \label{hlei.n} |
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395 | \end{equation} |
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396 | is multiplied by the value of $S_e$ as predicted by the Bethe-Bloch formula |
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397 | for $T > T_{2}$. |
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398 | The parameter $B$ is determined for each element of the material in order to |
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399 | ensure continuity at $T_p=T_2$. The value of $B$ for different atoms is |
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400 | usually less than 0.01. |
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401 | |
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402 | For kinetic energies of $\alpha$-particles $T < 1~keV$ the model |
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403 | of free electron gas \cite{hlei.Lindhard} is used |
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404 | \begin{equation} |
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405 | S_e = A \sqrt{T}, |
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406 | \label{hlei.o} |
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407 | \end{equation} |
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408 | The parameter $A$ is defined for each atom by requiring the stopping power to be |
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409 | continuous at $T=1~keV$. |
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410 | |
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411 | |
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412 | \subsection{Effective charge of ions} |
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413 | |
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414 | For hadrons or ions |
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415 | the scaling relation can be written as |
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416 | \begin{equation} |
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417 | S_{ei}(T) = Z_{eff}^2\cdot S_{ep}(T_p), |
---|
418 | \label{hlei.sei} |
---|
419 | \end{equation} |
---|
420 | where $S_{ei}$ is the ion stopping power, |
---|
421 | $S_{ep}$ is the proton stopping power at the energy scaled |
---|
422 | according (\ref{hlei.e}), and |
---|
423 | $Z_{eff}$ is effective charge of the particle, which has to be used in |
---|
424 | all expressions in place of $Z_h$. |
---|
425 | For fast particles it is equal to the particle charge $Z_h$, |
---|
426 | but for slow ions it differs significantly because |
---|
427 | a slow ion |
---|
428 | picks up electrons from the medium. |
---|
429 | The ion effective charge is expressed via |
---|
430 | the ion charge $Z_h$ and the |
---|
431 | fractional effective charge of ion $\gamma_i$: |
---|
432 | \begin{equation} |
---|
433 | Z_{eff} = \gamma_i Z_h. |
---|
434 | \label{hlei.pp} |
---|
435 | \end{equation} |
---|
436 | |
---|
437 | For helium ions |
---|
438 | fractional effective charge |
---|
439 | is parameterised for all |
---|
440 | elements with good accuracy \cite{hlei.Ziegler85} according to: |
---|
441 | \begin{eqnarray} |
---|
442 | (\gamma_{He})^2 & = &\left (1-\exp\left [-\sum_{j=0}^5{C_jQ^j}\right ]\right) |
---|
443 | \left ( 1 + \frac{ 7 + 0.05 Z }{1000} \exp( -(7.6-Q)^2 ) \right )^2, |
---|
444 | \nonumber \\ |
---|
445 | Q & = & \max ( 0, \ln T_p) , |
---|
446 | \label{hlei.q} |
---|
447 | \end{eqnarray} |
---|
448 | where the coefficients $C_j$ are the same for all elements, and the |
---|
449 | helium ion kinetic energy is in $keV/amu$. |
---|
450 | |
---|
451 | |
---|
452 | The following expression is used for heavy ions \cite{hlei.BK}: |
---|
453 | \begin{equation} |
---|
454 | \gamma_i = \left ( q + \frac{1-q}{2} \left (\frac{v_0}{v_F} \right )^2 |
---|
455 | \ln {\left ( 1 + \Lambda^2 \right )} \right ) |
---|
456 | \left ( 1 + \frac{(0.18+0.0015Z)\exp(-(7.6-Q)^2)}{Z_i^2} \right ), |
---|
457 | \label{hlei.s} |
---|
458 | \end{equation} |
---|
459 | where $q$ is |
---|
460 | the fractional average charge of the ion, |
---|
461 | $v_0$ is the Bohr velocity, |
---|
462 | $v_F$ is the Fermi velocity of |
---|
463 | the electrons in the target medium, and $\Lambda$ is |
---|
464 | the term taking into account the screening effect. In Ref.~\cite{hlei.BK}, |
---|
465 | $\Lambda$ is estimated to be: |
---|
466 | \begin{equation} |
---|
467 | \Lambda = 10 \frac{v_F}{v_0} \frac{(1-q)^{2/3}}{Z_i^{1/3}(6+q)}. |
---|
468 | \label{hlei.t} |
---|
469 | \end{equation} |
---|
470 | The Fermi velocity of the medium is of the same order as the Bohr velocity, and |
---|
471 | its exact value depends on the detailed electronic structure of the medium. |
---|
472 | Experimental data on the Fermi velocity are taken from |
---|
473 | Ref.\cite{hlei.Ziegler85}. |
---|
474 | The expression for the fractional average charge of the ion is the following: |
---|
475 | \begin{equation} |
---|
476 | q = [1 -\exp(0.803y^{0.3}-1.3167y^{0.6}-0.38157y-0.008983y^2)], |
---|
477 | \label{hlei.u} |
---|
478 | \end{equation} |
---|
479 | where $y$ is a parameter that depends on the ion velocity $v_i$ |
---|
480 | \begin{equation} |
---|
481 | y = \frac{v_i}{v_0Z^{2/3}} \left ( 1 +\frac {v_F^2}{5v_i^2} \right ). |
---|
482 | \label{hlei.v} |
---|
483 | \end{equation} |
---|
484 | |
---|
485 | The parametrisation described in this chapter is only valid |
---|
486 | if the reduced kinetic energy of the ion is higher than the lower limit |
---|
487 | of the energy: |
---|
488 | \begin{equation} |
---|
489 | T_p > \max \left ( 3.25~keV, \frac{25~keV}{Z^{2/3}} \right ). |
---|
490 | \label{hlei.x} |
---|
491 | \end{equation} |
---|
492 | If the ion energy is lower, then the free electron gas model (\ref{hlei.o}) |
---|
493 | is used to calculate the stopping power. |
---|
494 | |
---|
495 | |
---|
496 | \subsection{Energy losses of slow negative particles} |
---|
497 | |
---|
498 | At low energies, e.g. below a few MeV for protons/antiprotons, the |
---|
499 | Bethe-Bloch formula is no longer accurate in describing the energy |
---|
500 | loss of charged hadrons and higher $Z$ terms should be taken in |
---|
501 | account. |
---|
502 | Odd terms in $Z$ lead to a significant difference between energy |
---|
503 | loss of positively and negatively charged particles. |
---|
504 | The energy loss of negative hadrons is scaled from that |
---|
505 | of antiprotons. |
---|
506 | The antiproton energy loss is calculated in the following way: |
---|
507 | \begin{itemize} |
---|
508 | \item |
---|
509 | if the material is elemental, the quantum harmonic oscillator model is used, as |
---|
510 | described in \cite{hlei.sigmund} and references therein. |
---|
511 | The lower limit of applicability of the model is chosen for all |
---|
512 | materials at $50~keV$. Below this value stopping power is set to constant |
---|
513 | equal to the $dE/dx$ at $50~keV$. |
---|
514 | \item |
---|
515 | if the material is not elemental, the energy loss is calculated |
---|
516 | down to $500~keV$ using the Barkas correction (\ref{hlei.n}) |
---|
517 | and at lower energies fitting the |
---|
518 | proton energy loss curve. |
---|
519 | \end{itemize} |
---|
520 | |
---|
521 | |
---|
522 | |
---|
523 | |
---|
524 | \subsection{Energy losses of hadrons in compounds} |
---|
525 | |
---|
526 | To obtain energy losses in |
---|
527 | a mixture or compound, |
---|
528 | the absorber can be thought of as made up of thin |
---|
529 | layers of pure elements with weights proportional to the electron |
---|
530 | density of the element in the absorber (Bragg's rule): |
---|
531 | \begin{equation} |
---|
532 | \frac{dE}{dx}=\sum_i{\left (\frac{dE}{dx} \right )_i}, |
---|
533 | \label{hlei.y} |
---|
534 | \end{equation} |
---|
535 | where the sum is taken over all elements of the absorber, $i$ is |
---|
536 | the number of the element, |
---|
537 | $(\frac{dE}{dx})_i$ is energy loss in the pure $i$-th element. |
---|
538 | |
---|
539 | Bragg's rule is very accurate for relativistic particles |
---|
540 | when the interaction of electrons with a nucleus is negligible. |
---|
541 | But at low energies the accuracy of Bragg's rule is limited |
---|
542 | because the energy loss to the electrons in any material |
---|
543 | depends on the detailed orbital |
---|
544 | and excitation structure of the material. |
---|
545 | In the description of Geant4 materials there is a special |
---|
546 | attribute: the chemical formula. |
---|
547 | It is used in the |
---|
548 | following way: |
---|
549 | \begin{itemize} |
---|
550 | \item |
---|
551 | if the data on the stopping power for a compound |
---|
552 | as a function of the proton kinetic energy |
---|
553 | is available (Table \ref{hlei.tab1}), then the |
---|
554 | direct parametrisation of the data for this material is performed; |
---|
555 | \item |
---|
556 | if the data on the stopping power for a compound |
---|
557 | is available for only one incident |
---|
558 | energy (Table \ref{hlei.tab2}), then |
---|
559 | the computation is |
---|
560 | performed based on Bragg's rule and the chemical |
---|
561 | factor for the compound is taken into account; |
---|
562 | \item |
---|
563 | if there are no data for the compound, the computation is |
---|
564 | performed based on Bragg's rule. |
---|
565 | \end{itemize} |
---|
566 | \noindent |
---|
567 | In the review \cite{hlei.Ziegler88} the parametrisation stopping |
---|
568 | power data are presented as |
---|
569 | \begin{equation} |
---|
570 | S_e(T_p)= S_{Bragg}(T_p)\left [1 + \frac{f(T_p)}{f(125~keV)} |
---|
571 | \left (\frac{S_{exp}(125~keV)}{S_{Bragg}(125~keV)}-1 \right ) \right ], |
---|
572 | \label{hlei.z} |
---|
573 | \end{equation} |
---|
574 | where $S_{exp}(125~keV)$ is the experimental value of the energy loss |
---|
575 | for the compound |
---|
576 | for $125~keV$ protons or the |
---|
577 | reduced experimental value for He ions, $S_{Bragg}(T_p)$ is |
---|
578 | a value of energy loss calculated according to Bragg's |
---|
579 | rule, and $f(T_p)$ is a universal function, which describes |
---|
580 | the disappearance of deviations from Bragg's rule |
---|
581 | for higher kinetic energies according to: |
---|
582 | \begin{equation} |
---|
583 | f(T_p)=\frac{1}{1+\exp \left [1.48(\frac{\beta(T_p)} |
---|
584 | {\beta(25~keV)}-7.0) \right ]}, |
---|
585 | \label{hlei.fun} |
---|
586 | \end{equation} |
---|
587 | where $\beta(T_p)$ is the relative velocity of the proton with |
---|
588 | kinetic energy $T_p$. |
---|
589 | |
---|
590 | |
---|
591 | \begin{table*} |
---|
592 | \caption{The list of chemical formulae of compounds for which |
---|
593 | parametrisation of stopping power as a function |
---|
594 | of kinetic energy is in Ref.\cite{hlei.ICRU49}.} |
---|
595 | %\vspace {2pt} |
---|
596 | \label{hlei.tab1} |
---|
597 | \begin{center} |
---|
598 | \begin{tabular}{|l|l|} |
---|
599 | \hline |
---|
600 | Number & Chemical formula \\ |
---|
601 | \hline |
---|
602 | 1. & AlO \\ |
---|
603 | 2. & C\_2O \\ |
---|
604 | 3. & CH\_4 \\ |
---|
605 | 4. & (C\_2H\_4)\_N-Polyethylene \\ |
---|
606 | 5. & (C\_2H\_4)\_N-Polypropylene \\ |
---|
607 | 6. & (C\_8H\_8)\_N \\ |
---|
608 | 7. & C\_3H\_8 \\ |
---|
609 | 8. & SiO\_2 \\ |
---|
610 | 9. & H\_2O \\ |
---|
611 | 10. & H\_2O-Gas \\ |
---|
612 | 11. & Graphite \\ |
---|
613 | \hline |
---|
614 | \end{tabular} |
---|
615 | \end{center} |
---|
616 | \end{table*} |
---|
617 | |
---|
618 | \begin{table*} |
---|
619 | \caption{The list of chemical formulae of compounds for which |
---|
620 | the {\it chemical factor} is calculated from the data |
---|
621 | of Ref.\cite{hlei.Ziegler88}.} |
---|
622 | %\vspace {2pt} |
---|
623 | \label{hlei.tab2} |
---|
624 | \begin{center} |
---|
625 | \begin{tabular}{|l|l|l|l|} |
---|
626 | \hline |
---|
627 | Number & Chemical formula & Number & Chemical formula \\ |
---|
628 | \hline |
---|
629 | 1. & H\_2O & 28. & C\_2H\_6 \\ |
---|
630 | 2. & C\_2H\_4O & 29. & C\_2F\_6 \\ |
---|
631 | 3. & C\_3H\_6O & 30. & C\_2H\_6O \\ |
---|
632 | 4. & C\_2H\_2 & 31. & C\_3H\_6O \\ |
---|
633 | 5. & C\_H\_3OH & 32. & C\_4H\_10O \\ |
---|
634 | 6. & C\_2H\_5OH & 33. & C\_2H\_4 \\ |
---|
635 | 7. & C\_3H\_7OH & 34. & C\_2H\_4O \\ |
---|
636 | 8. & C\_3H\_4 & 35. & C\_2H\_4S \\ |
---|
637 | 9. & NH\_3 & 36. & SH\_2 \\ |
---|
638 | 10. & C\_14H\_10 & 37. & CH\_4 \\ |
---|
639 | 11. & C\_6H\_6 & 38. & CCLF\_3 \\ |
---|
640 | 12. & C\_4H\_10 & 39. & CCl\_2F\_2 \\ |
---|
641 | 13. & C\_4H\_6 & 40. & CHCl\_2F \\ |
---|
642 | 14. & C\_4H\_8O & 41. & (CH\_3)\_2S \\ |
---|
643 | 15. & CCl\_4 & 42. & N\_2O \\ |
---|
644 | 16. & CF\_4 & 43. & C\_5H\_10O \\ |
---|
645 | 17. & C\_6H\_8 & 44. & C\_8H\_6 \\ |
---|
646 | 18. & C\_6H\_12 & 45. & (CH\_2)\_N \\ |
---|
647 | 19. & C\_6H\_10O & 46. & (C\_3H\_6)\_N \\ |
---|
648 | 20. & C\_6H\_10 & 47. & (C\_8H\_8)\_N \\ |
---|
649 | 21. & C\_8H\_16 & 48. & C\_3H\_8 \\ |
---|
650 | 22. & C\_5H\_10 & 49. & C\_3H\_6-Propylene \\ |
---|
651 | 23. & C\_5H\_8 & 50. & C\_3H\_6O \\ |
---|
652 | 24. & C\_3H\_6-Cyclopropane & 51. & C\_3H\_6S \\ |
---|
653 | 25. & C\_2H\_4F\_2 & 52. & C\_4H\_4S \\ |
---|
654 | 26. & C\_2H\_2F\_2 & 53. & C\_7H\_8 \\ |
---|
655 | 27. & C\_4H\_8O\_2 & & \\ |
---|
656 | \hline |
---|
657 | \end{tabular} |
---|
658 | \end{center} |
---|
659 | \end{table*} |
---|
660 | |
---|
661 | |
---|
662 | \subsection{Nuclear stopping powers} |
---|
663 | |
---|
664 | Low energy ions transfer their energy not only to electrons of a medium |
---|
665 | but also to the nuclei of the medium due to the elastic Coulomb |
---|
666 | collisions. |
---|
667 | This contribution to the energy loss is called {\it |
---|
668 | nuclear stopping power}. |
---|
669 | It is parametrised \cite{hlei.Ziegler774,hlei.Ziegler85,hlei.ICRU49} |
---|
670 | using a universal parameterisation for reduced |
---|
671 | ion energy, $\epsilon$, which depends on ion parameters and on |
---|
672 | the charge, $Z_t$, and the mass, $M_t$, of the target nucleus: |
---|
673 | \begin{equation} |
---|
674 | \epsilon = \frac{32.536TM_t}{Z_{eff}Z_t(M+M_t) |
---|
675 | \sqrt{Z_{eff}^{0.23}+Z_t^{0.23}}}. |
---|
676 | \label{hlei.ep} |
---|
677 | \end{equation} |
---|
678 | The universal reduced nuclear stopping power, $s_n$, is determined |
---|
679 | by J.~Moliere in the framework of Thomas-Fermi potential \cite{hlei.mol}. |
---|
680 | The corresponding tabulation from Ref.\cite{hlei.ICRU49} |
---|
681 | is implemented. |
---|
682 | To transform the value of |
---|
683 | nuclear stopping power from reduced units to |
---|
684 | $[eV/10^{15}atoms/cm^2]$ the following formula is used: |
---|
685 | \begin{equation} |
---|
686 | S_n = s_n \frac{8.462Z_iZ_tM_i}{(M_i+M_t)\sqrt{Z_i^{0.23}+Z_t^{0.23}}}. |
---|
687 | \label{hlei.re} |
---|
688 | \end{equation} |
---|
689 | The effect of nuclear stopping power is very small at high energies, but |
---|
690 | it is of the same order of magnitude as electronic stopping power |
---|
691 | for very slow ions (e.g. for protons, $T_p < 1 keV$). |
---|
692 | |
---|
693 | \subsection{Fluctuations of energy losses of hadrons} |
---|
694 | |
---|
695 | The total continuous energy loss of charged particles is a stochastic |
---|
696 | quantity with a distribution described in terms of a straggling function. |
---|
697 | The straggling is partially taken into account by the simulation |
---|
698 | of energy loss by the production of $\delta$-electrons with energy |
---|
699 | $T > T_c$. However, continuous energy loss also has fluctuations. Hence |
---|
700 | in the current GEANT4 implementation two different models of fluctuations |
---|
701 | are applied depending on the value of the parameter $\kappa$ which is the |
---|
702 | lower limit of the number of interactions of the particle in the step. |
---|
703 | The default value chosen is $\kappa = 10$. To select a model for thick |
---|
704 | absorbers the following boundary conditions are used: |
---|
705 | \begin{equation} |
---|
706 | \Delta E > T_c\kappa)\;\; or \;\; T_c < I\kappa, |
---|
707 | \label{le_cond} |
---|
708 | \end{equation} |
---|
709 | where $\Delta E$ is the mean continuous energy loss in a track segment of |
---|
710 | length $s$, $T_c$ is the cut kinetic energy of $\delta$-electrons, and $I$ |
---|
711 | is the average ionisation potential of the atom. |
---|
712 | |
---|
713 | For long path lengths the straggling function |
---|
714 | approaches the Gaussian distribution with Bohr's variance \cite{hlei.ICRU49}: |
---|
715 | \begin{equation} |
---|
716 | \Omega^2 = K N_{el}\frac{Z_h^2}{\beta^2} T_c s f |
---|
717 | \left(1 - \frac{\beta^2}{2} \right), |
---|
718 | \label{sig} |
---|
719 | \end{equation} |
---|
720 | where $f$ is a screening factor, which is equal to unity for fast particles, |
---|
721 | whereas for slow positively charged |
---|
722 | ions with $\beta^2 < 3Z (v_0/c)^2$ |
---|
723 | $f = a + b/Z^2_{eff}$, where parameters $a$ and $b$ |
---|
724 | are parametrised for all atoms \cite{hlei.Yang,hlei.Chu}. |
---|
725 | |
---|
726 | For short path lengths, when the condition \ref{le_cond} is not satisfied, |
---|
727 | the model described in the charter \ref{gen_fluctuations} is applied. |
---|
728 | |
---|
729 | \subsection{Sampling} |
---|
730 | |
---|
731 | At each step for a charged hadron or ion in an absorber, |
---|
732 | the step limit is calculated using range tables |
---|
733 | for protons or antiprotons depending on the particle charge. |
---|
734 | If the reduced particle energy $T_p < T_2$ the step limit is |
---|
735 | forced to be not longer than $\alpha R(T_2)$, where $R(T_2)$ |
---|
736 | is the range of the particle with the reduced energy $T_2$, |
---|
737 | $\alpha$ is an arbitrary coefficient, which is currently set to 0.05 |
---|
738 | in order to provide at least 20 steps for particles |
---|
739 | in the Bragg peak energy range. |
---|
740 | \noindent |
---|
741 | In each step continuous energy loss of the particle |
---|
742 | is calculated using loss tables for protons or antiprotons |
---|
743 | for $T_p > T_2$. For lower energies, continuous energy loss |
---|
744 | is calculated using the effective charge approach, chemical |
---|
745 | factors, and nuclear stopping powers. |
---|
746 | \noindent |
---|
747 | If the step of the particle is limited by the ionisation process |
---|
748 | the sampling of $\delta$-electron production is performed. |
---|
749 | (A short overview of the method is given in Chapter \ref{secmessel}.) \\ |
---|
750 | Apart from the normalisation, the cross-section |
---|
751 | (\ref{hlei.bbb}) can be written as : |
---|
752 | \begin{eqnarray} |
---|
753 | \frac{d\sigma}{dT} \sim f(T) \ g(T) &with& T \in [T_{c}, \ T_{max}] |
---|
754 | \end{eqnarray} |
---|
755 | with : |
---|
756 | \begin{eqnarray*} |
---|
757 | f(T) &=& \left(\frac{1}{T_{c}} -\frac{1}{T_{max}}\right) |
---|
758 | \frac{1}{T^2} \\ |
---|
759 | g(T) &=& 1 - \beta^2\frac{T}{T_{max}} + S(T), |
---|
760 | \end{eqnarray*} |
---|
761 | where $S(T)$ is a spin dependent term (\ref{hlei.bbb}). |
---|
762 | For a spin-0 particle this term is zero, for |
---|
763 | a spin-$\frac{1}{2}$ particle $S(T)=T^2/2E^2$, |
---|
764 | whilst for spin-1 the expression is more complicated. |
---|
765 | \\ |
---|
766 | The energy, $T$, is sampled by : |
---|
767 | \begin{enumerate} |
---|
768 | \item Sample $T$ from $f(T)$. |
---|
769 | \item Calculate the rejection function $g(T)$ and accept the |
---|
770 | sampled $T$ with a probability of $g(T)$. |
---|
771 | \end{enumerate} |
---|
772 | After the successful sampling of the energy, the polar angles of the |
---|
773 | emitted electron are generated with respect to the direction of the |
---|
774 | incident particle. The azimuthal angle, $\phi$, is generated isotropically; |
---|
775 | the polar angle $\theta$ is calculated from the energy momentum conservation. |
---|
776 | This information is used to calculate the energy and momentum of both |
---|
777 | particles and to transform them into the {\it global} coordinate system. |
---|
778 | |
---|
779 | \subsection{PIXE} |
---|
780 | PIXE is simulated by calculating cross-sections according to |
---|
781 | \cite{hlei.Gryzinski1} and \cite{hlei.Gryzinski2} to identify the primary |
---|
782 | ionised shell, and generating the subsequent atomic relaxation as described |
---|
783 | in \ref{relax}. Sampling of excitations is carried out for both the |
---|
784 | continuous and the discrete parts of the process. |
---|
785 | |
---|
786 | |
---|
787 | \subsection{ICRU 73-based energy loss model} |
---|
788 | The ICRU 73 \cite{hlei.ICRU73} report contains stopping power tables |
---|
789 | for ions with atomic numbers 3--18 and 26, covering a range of different |
---|
790 | elemental and compound target materials. The stopping powers derive from |
---|
791 | calculations with the PASS code \cite{hlei.sigm02}, which implements the |
---|
792 | binary stopping theory described in \cite{hlei.sigm02,hlei.sigm00}. Tables |
---|
793 | in ICRU 73 extend over an energy range up to 1 GeV/nucleon. All stopping |
---|
794 | powers were incorporated into Geant4 and are available through a |
---|
795 | parameterisation model ({\tt G4IonParametrisedLossModel}). For a few |
---|
796 | materials revised stopping powers were included (water, water vapor, nylon type |
---|
797 | 6 and 6/6 from P. Sigmund et al \cite{hlei.sigm09a} and copper from P. Sigmund |
---|
798 | \cite{hlei.sigm09b}), which replace the corresponding tables of the original |
---|
799 | ICRU 73 report. |
---|
800 | |
---|
801 | To account for secondary electron production above $T_{c}$, the continuous |
---|
802 | energy loss per unit path length is calculated according to |
---|
803 | \begin{equation} |
---|
804 | \label{hlei.rstp} |
---|
805 | \frac{dE}{dx}\bigg|_{T<T_C} = \bigg(\frac{dE}{dx}\bigg)_{ICRU73} - |
---|
806 | \bigg(\frac{dE}{dx}\bigg)_{\delta} |
---|
807 | \end{equation} |
---|
808 | where $(dE/dx)_{ICRU73}$ refers to stopping powers obtained by interpolating |
---|
809 | ICRU 73 tables and $(dE/dx)_{\delta}$ is the mean energy transferred to |
---|
810 | $\delta$-electrons per path length given by |
---|
811 | \begin{equation} |
---|
812 | \bigg(\frac{dE}{dx}\bigg)_{\delta} = \sum_{i} n_{at,i} \int_{T_c}^{T_{max}} |
---|
813 | \frac{d\sigma_i(T)}{dT} T dT |
---|
814 | \label{} |
---|
815 | \end{equation} |
---|
816 | where the index $i$ runs over all elements composing the material, $n_{at,i}$ |
---|
817 | is the number of atoms of the element $i$ per volume, $T_{max}$ is the maximum |
---|
818 | energy transferable to an electron according to formula (\ref{hlei.a1}) and |
---|
819 | $d\sigma_i/dT$ specifies the differential cross section per atom for producing |
---|
820 | an $\delta$-electron following equation (\ref{hlei.bbb}). |
---|
821 | |
---|
822 | For compound targets not considered in the ICRU 73 report, the first term on |
---|
823 | the rightern side in equation (\ref{hlei.rstp}) is computed by applying Bragg's |
---|
824 | additivity rule \cite{hlei.ICRU49} if tables for all elemental components are |
---|
825 | available in ICRU 73. |
---|
826 | |
---|
827 | |
---|
828 | |
---|
829 | \subsection{Status of this document} |
---|
830 | |
---|
831 | \noindent |
---|
832 | 21.11.2000 Created by V.Ivanchenko \\ |
---|
833 | 30.05.2001 Modified by V.Ivanchenko \\ |
---|
834 | 23.11.2001 Modified by M.G. Pia to add PIXE section. \\ |
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835 | 19.01.2002 Minor corrections (mma) \\ |
---|
836 | 13.05.2002 Minor corrections (V.Ivanchenko) \\ |
---|
837 | 28.08.2002 Minor corrections (V.Ivanchenko) \\ |
---|
838 | 11.12.2009 Modified by A. Lechner to add ICRU 73 section |
---|
839 | |
---|
840 | \begin{latexonly} |
---|
841 | |
---|
842 | \begin{thebibliography}{599} |
---|
843 | |
---|
844 | \bibitem{hlei.prepHadr}V.N.~Ivanchenko et al., GEANT4 Simulation |
---|
845 | of |
---|
846 | Energy Losses of Slow Hadrons, CERN-99-121, INFN/AE-99/20, (September 1999). |
---|
847 | \bibitem{hlei.prepIon}S.~Giani et al., GEANT4 Simulation |
---|
848 | of |
---|
849 | Energy Losses of Ions, CERN-99-300, INFN/AE-99/21, (November 1999). |
---|
850 | \bibitem{hlei.pdg} D.E.~Groom et al., Eur. |
---|
851 | Phys. Jour. C15 (2000) 1. |
---|
852 | \bibitem{hlei.rossi52} B.~Rossi, High Energy |
---|
853 | Particles, Pretice-Hall, Inc., Englewood Cliffs, NJ, 1952. |
---|
854 | \bibitem{hlei.bethe}H.~Bethe, Ann. Phys. 5 (1930) 325. |
---|
855 | \bibitem{hlei.ICRU37} (A.~Allisy et al), |
---|
856 | Stopping Powers for Electrons and Positrons, |
---|
857 | ICRU Report 37, 1984. |
---|
858 | \bibitem{hlei.sternheimer} |
---|
859 | R.M.~Sternheimer. Phys.Rev. B3 (1971) 3681. |
---|
860 | \bibitem{hlei.bark62} |
---|
861 | W.H.~Barkas. Technical Report 10292,UCRL, August 1962. |
---|
862 | \bibitem{hlei.bark56} |
---|
863 | W.H.~Barkas, W.~Birnbaum, F.M.~Smith, Phys. Rev. |
---|
864 | 101 (1956) 778. |
---|
865 | \bibitem{hlei.arb72} |
---|
866 | J.C.~Ashley, R.H.~Ritchie and W.~Brandt, |
---|
867 | Phys. Rev. B5 (1972) 1. |
---|
868 | \bibitem{hlei.bloch}F.~Bloch, Ann. Phys. 16 (1933) 285. |
---|
869 | \bibitem{hlei.Lindhard} |
---|
870 | J.~Linhard and A.~Winther, Mat. Fys. Medd. Dan. Vid. Selsk. |
---|
871 | 34, No 10 (1963). |
---|
872 | \bibitem{hlei.Ziegler771}H.H.~Andersen and J.F.~Ziegler, |
---|
873 | The Stopping |
---|
874 | and Ranges of Ions in Matter. Vol.3, Pergamon Press, 1977. |
---|
875 | \bibitem{hlei.ICRU49}ICRU (A.~Allisy et al), |
---|
876 | Stopping Powers and Ranges for Protons and Alpha |
---|
877 | Particles, |
---|
878 | ICRU Report 49, 1993. |
---|
879 | \bibitem{hlei.Ziegler774}J.F.~Ziegler, The Stopping |
---|
880 | and Ranges of Ions in Matter. Vol.4, Pergamon Press, 1977. |
---|
881 | \bibitem{hlei.Ziegler85}J.F.~Ziegler, J.P.~Biersack, U |
---|
882 | .~Littmark, The Stopping |
---|
883 | and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985. |
---|
884 | \bibitem{hlei.BK} |
---|
885 | W.~Brandt and M.~Kitagawa, Phys. Rev. B25 (1982) 5631. |
---|
886 | \bibitem{hlei.sigmund} |
---|
887 | P.~Sigmund, Nucl. Instr. and Meth. |
---|
888 | B85 (1994) 541. |
---|
889 | \bibitem{hlei.Ziegler88} J.F.~Ziegler and |
---|
890 | J.M.~Manoyan, Nucl. Instr. and Meth. |
---|
891 | B35 (1988) 215. |
---|
892 | \bibitem{hlei.mol}G.~Moliere, |
---|
893 | Theorie der Streuung schneller geladener Teilchen I; |
---|
894 | Einzelstreuungam abbgeschirmten Coulomb-Feld, Z. f. Naturforsch, A2 |
---|
895 | (1947) 133. |
---|
896 | \bibitem{hlei.GEANT3} GEANT3 manual, |
---|
897 | CERN Program Library Long Writeup |
---|
898 | W5013 (October 1994). |
---|
899 | \bibitem{hlei.Yang} Q.~Yang, |
---|
900 | D.J.~O'Connor, Z.~Wang, Nucl. Instr. and Meth. |
---|
901 | B61 (1991) 149. |
---|
902 | \bibitem{hlei.Chu} W.K.~Chu, in: Ion Beam Handbook for |
---|
903 | Material Analysis, edt. J.W.~Mayer and E.~Rimini, |
---|
904 | Academic Press, NY, 1977. |
---|
905 | \bibitem{hlei.Gryzinski1} M. Gryzinski, Phys. Rev. A 135 (1965) 305. |
---|
906 | \bibitem{hlei.Gryzinski2} M. Gryzinski, Phys. Rev. A 138 (1965) 322. |
---|
907 | \bibitem{hlei.ICRU73} |
---|
908 | Stopping of Ions Heavier Than Helium, |
---|
909 | ICRU Report 73, Oxford University Press (2005). |
---|
910 | \bibitem{hlei.sigm02} |
---|
911 | P.~Sigmund and A.~Schinner, |
---|
912 | Nucl. Instr. Meth. in Phys. Res. B 195 (2002) 64. |
---|
913 | \bibitem{hlei.sigm00} |
---|
914 | P.~Sigmund and A.~Schinner, |
---|
915 | Eur. Phys. J. D 12 (2000) 425. |
---|
916 | \bibitem{hlei.sigm09a} |
---|
917 | P.~Sigmund, A.~Schinner and H.~Paul, |
---|
918 | Errata and Addenda for ICRU Report 73, Stopping of Ions Heavier |
---|
919 | than Helium (2009). |
---|
920 | \bibitem{hlei.sigm09b} |
---|
921 | Personal communication with P.~Sigmund (2009). |
---|
922 | \end{thebibliography} |
---|
923 | |
---|
924 | \end{latexonly} |
---|
925 | |
---|
926 | \begin{htmlonly} |
---|
927 | |
---|
928 | \subsection{Bibliography} |
---|
929 | |
---|
930 | \begin{enumerate} |
---|
931 | \item V.N.~Ivanchenko et al., GEANT4 Simulation of |
---|
932 | Energy Losses of Slow Hadrons, CERN-99-121, INFN/AE-99/20, (September 1999). |
---|
933 | \item S.~Giani et al., GEANT4 Simulation of |
---|
934 | Energy Losses of Ions, CERN-99-300, INFN/AE-99/21, (November 1999). |
---|
935 | \item D.E.~Groom et al., Eur. |
---|
936 | Phys. Jour. C15 (2000) 1. |
---|
937 | \item B.~Rossi, High Energy |
---|
938 | Particles, Pretice-Hall, Inc., Englewood Cliffs, NJ, 1952. |
---|
939 | \item H.~Bethe, Ann. Phys. 5 (1930) 325. |
---|
940 | \item (A.~Allisy et al), |
---|
941 | Stopping Powers for Electrons and Positrons, |
---|
942 | ICRU Report 37, 1984. |
---|
943 | \item |
---|
944 | R.M.~Sternheimer. Phys.Rev. B3 (1971) 3681. |
---|
945 | \item |
---|
946 | W.H.~Barkas. Technical Report 10292,UCRL, August 1962. |
---|
947 | \item |
---|
948 | W.H.~Barkas, W.~Birnbaum, F.M.~Smith, Phys. Rev. |
---|
949 | 101 (1956) 778. |
---|
950 | \item |
---|
951 | J.C.~Ashley, R.H.~Ritchie and W.~Brandt, |
---|
952 | Phys. Rev. B5 (1972) 1. |
---|
953 | \item F.~Bloch, Ann. Phys. 16 (1933) 285. |
---|
954 | \item |
---|
955 | J.~Linhard and A.~Winther, Mat. Fys. Medd. Dan. Vid. Selsk. |
---|
956 | 34, No 10 (1963). |
---|
957 | \item H.H.~Andersen and J.F.~Ziegler, |
---|
958 | The Stopping |
---|
959 | and Ranges of Ions in Matter. Vol.3, Pergamon Press, 1977. |
---|
960 | \item ICRU (A.~Allisy et al), |
---|
961 | Stopping Powers and Ranges for Protons and Alpha |
---|
962 | Particles, |
---|
963 | ICRU Report 49, 1993. |
---|
964 | \item J.F.~Ziegler, The Stopping |
---|
965 | and Ranges of Ions in Matter. Vol.4, Pergamon Press, 1977. |
---|
966 | \item J.F.~Ziegler, J.P.~Biersack, U |
---|
967 | .~Littmark, The Stopping |
---|
968 | and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985. |
---|
969 | \item |
---|
970 | W.~Brandt and M.~Kitagawa, Phys. Rev. B25 (1982) 5631. |
---|
971 | \item |
---|
972 | P.~Sigmund, Nucl. Instr. and Meth. |
---|
973 | B85 (1994) 541. |
---|
974 | \item J.F.~Ziegler and |
---|
975 | J.M.~Manoyan, Nucl. Instr. and Meth. |
---|
976 | B35 (1988) 215. |
---|
977 | \item G.~Moliere, |
---|
978 | Theorie der Streuung schneller geladener Teilchen I; |
---|
979 | Einzelstreuungam abbgeschirmten Coulomb-Feld, Z. f. Naturforsch, A2 |
---|
980 | (1947) 133. |
---|
981 | \item GEANT3 manual, |
---|
982 | CERN Program Library Long Writeup |
---|
983 | W5013 (October 1994). |
---|
984 | \item Q.~Yang, |
---|
985 | D.J.~O'Connor, Z.~Wang, Nucl. Instr. and Meth. |
---|
986 | B61 (1991) 149. |
---|
987 | \item W.K.~Chu, in: Ion Beam Handbook for |
---|
988 | Material Analysis, edt. J.W.~Mayer and E.~Rimini, |
---|
989 | Academic Press, NY, 1977. |
---|
990 | \item M. Gryzinski, Phys. Rev. A 135 (1965) 305. |
---|
991 | \item M. Gryzinski, Phys. Rev. A 138 (1965) 322. |
---|
992 | \item |
---|
993 | Stopping of Ions Heavier Than Helium, |
---|
994 | ICRU Report 73, Oxford University Press (2005). |
---|
995 | \item |
---|
996 | P.~Sigmund and A.~Schinner, |
---|
997 | Nucl. Instr. Meth. in Phys. Res. B 195 (2002) 64. |
---|
998 | \item |
---|
999 | P.~Sigmund and A.~Schinner, |
---|
1000 | Eur. Phys. J. D 12 (2000) 425. |
---|
1001 | \item |
---|
1002 | P.~Sigmund, A.~Schinner and H.~Paul, |
---|
1003 | Errata and Addenda for ICRU Report 73, Stopping of Ions Heavier than Helium (2009). |
---|
1004 | \item |
---|
1005 | Personal communication with P.~Sigmund (2009). |
---|
1006 | |
---|
1007 | \end{enumerate} |
---|
1008 | |
---|
1009 | \end{htmlonly} |
---|
1010 | |
---|
1011 | |
---|