1 | \section[Ionization]{Hadron and Ion Ionization} \label{hion} |
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2 | |
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3 | \subsection{Method} |
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4 | |
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5 | The class {\it G4hIonisation} provides the continuous energy loss due to |
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6 | ionization and simulates the 'discrete' part of the ionization, that is, |
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7 | delta rays produced by charged hadrons. The class {\it G4ionIonisation} is |
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8 | intended for the simulation of energy loss by positive ions with change greater than unit. |
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9 | Inside these classes the following models are used: |
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10 | \begin{itemize} |
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11 | \item |
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12 | {\it G4BetherBlochModel} (valid for protons with $T > 2\; MeV$) |
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13 | \item |
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14 | {\it G4BraggModel} (valid for protons with $T < 2\; MeV$) |
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15 | \item |
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16 | {\it G4BraggIonModel} (valid for protons with $T < 2\; MeV$) |
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17 | \end{itemize} |
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18 | The scaling relation (\ref{enloss.sc}) is a basic conception for the description |
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19 | of ionization of heavy charged particles. It is used both in energy loss |
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20 | calculation and in determination of the validity range of models. Namely the |
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21 | $T_p = 2 MeV$ limit for protons is scaled for a particle with mass $M_i$ |
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22 | by the ratio of the particle mass to the proton mass $T_i = T_p M_p/M_i$. |
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23 | |
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24 | For all ionization models the value of the maximum energy |
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25 | transferable to a free electron $T_{max}$ is given by the following relation \cite{hion.pdg}: |
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26 | \begin{equation} |
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27 | \label{hion.c} |
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28 | T_{max} =\frac{2m_ec^2(\gamma^2 -1)}{1+2\gamma (m_e/M)+(m_e/M)^2 }, |
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29 | \end{equation} |
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30 | where $m_e$ is the electron mass and $M$ is the mass of the incident particle. |
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31 | The method of calculation of the continuous energy loss and the total |
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32 | cross-section are explained below. |
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33 | |
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34 | \subsection{Continuous Energy Loss} |
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35 | |
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36 | The integration of \ref{comion.a} leads to the Bethe-Bloch restricted energy |
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37 | loss ($T < T_{cut}$ formula \cite{hion.pdg}, which is modified taken into |
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38 | account various corrections \cite{hion.ahlen}: |
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39 | \begin{equation} |
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40 | \label{hion.d} |
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41 | \frac{dE}{dx} = |
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42 | 2 \pi r_e^2 mc^2 n_{el} \frac{z^2}{\beta^2} |
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43 | \left [\ln \left(\frac{2mc^2 \beta^2 \gamma^2 T_{up}} {I^2} \right) |
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44 | - \beta^2 \left( 1 + \frac{T_{up}}{T_{max}} \right) |
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45 | - \delta - \frac{2C_e}{Z} + F\right ] |
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46 | \end{equation} |
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47 | where |
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48 | \[ |
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49 | \begin{array}{ll} |
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50 | r_e & \mbox{classical electron radius:} |
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51 | \quad e^2/(4 \pi \epsilon_0 mc^2 ) \\ |
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52 | mc^2 & \mbox{mass-energy of the electron} \\ |
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53 | n_{el} & \mbox{electrons density in the material} \\ |
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54 | I & \mbox{mean excitation energy in the material}\\ |
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55 | Z & \mbox{atomic number of the material} \\ |
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56 | $z$ & \mbox{charge of the hadron in units of the electron change} \\ |
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57 | \gamma & \mbox{$E/mc^2$} \\ |
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58 | \beta^2 & 1-(1/\gamma^2) \\ |
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59 | T_{up} & \min(T_{cut},T_{max}) \\ |
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60 | \delta & \mbox{density effect function} \\ |
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61 | C_e & \mbox{shell correction function} \\ |
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62 | F & \mbox{high order corrections} |
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63 | \end{array} |
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64 | \] |
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65 | In a single element the electron density is |
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66 | $$ n_{el} = Z \: n_{at} = Z \: \frac{\mathcal{N}_{av} \rho}{A} $$ |
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67 | ($\mathcal{N}_{av}$: Avogadro number, $\rho$: density of the material, |
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68 | $A$: mass of a mole). In a compound material |
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69 | $$ |
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70 | n_{el} = \sum_i Z_i \: n_{ati} |
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71 | = \sum_i Z_i \: \frac{\mathcal{N}_{av} w_i \rho}{A_i} . |
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72 | $$ |
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73 | $w_i$ is the proportion by mass of the $i^{th}$ element, with molar mass $A_i$. |
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74 | |
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75 | |
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76 | The mean excitation energy $I$ for all elements is tabulated according to |
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77 | the ICRU recommended values \cite{hion.ICRU37}. |
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78 | |
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79 | \subsubsection{Shell Correction} |
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80 | |
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81 | $2C_e/Z$ is the so-called {\it shell correction term} which accounts for the |
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82 | fact of interaction of atomic electrons with atomic nucleus. This term more visible |
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83 | at low energies and for heavy atoms. The classical |
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84 | expression for the term \cite{hion.ICRU49} is used |
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85 | \begin{equation} |
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86 | \label{hion.dh} |
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87 | C = \sum{C_{\nu}(\theta_{\nu},\eta_{\nu})}, \;\; \nu=K,L,M,..., |
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88 | \; \theta=\frac{J_{\nu}}{\epsilon_{\nu}}, \;\; \eta_{\nu}=\frac{\beta^2}{\alpha^2 Z^2_{\nu}}, |
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89 | \end{equation} |
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90 | where $\alpha$ is the fine structure constant, $\beta$ is the hadron velocity, |
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91 | $J_{\nu}$ is the ionisation energy of the shell $\nu$, $\epsilon_{\nu}$ is |
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92 | Bohr ionisation energy of the shell $\nu$, $Z_{\nu}$ is the effective charge of the |
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93 | shell $\nu$. |
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94 | First terms $C_K$ and $C_L$ can be analytically computed in using an assumption |
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95 | non-relativistic hydrogenic wave functions \cite{hion.37,hion.38}. |
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96 | The results \cite{hion.39} of tabulation of these computations in the interval of parameters |
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97 | $\eta_{\nu} = 0.005\div 10$ and $\theta_{\nu}=0.25 \div 0.95$ are used directly. |
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98 | For higher values of $\eta_{\nu}$ the parameterization \cite{hion.39} |
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99 | is applied: |
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100 | \begin{equation} |
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101 | C_{\nu} = \frac{K_1}{\eta} + \frac{K_2}{\eta^2} + \frac{K_3}{\eta^3}, |
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102 | \end{equation} |
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103 | where coefficients $K_i$ provide smooth shape of the function. |
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104 | The effective nuclear charge for the $L$-shell can be reproduced as $Z_L = Z - d$, $d$ |
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105 | is a parameter shown in Table \ref{hion.t}. |
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106 | \begin{table}[hbt] |
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107 | \begin{centering} |
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108 | \begin{tabular}{|c|c|c|c|c|c|c|c|c|} |
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109 | \hline |
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110 | $Z$ & 3 & 4 & 5 & 6 & 7 & 8 & 9 & $>$9\\ \hline |
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111 | $d$ & 1.72 & 2.09 & 2.48 & 2.82 & 3.16 & 3.53 & 3.84 & 4.15\\ \hline |
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112 | \end{tabular} |
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113 | \caption{Effective nuclear charge for the $L$-shell \cite{hion.ICRU49}.} |
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114 | \label{hion.th} |
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115 | \end{centering} |
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116 | \end{table} |
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117 | For outer |
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118 | shells the calculations are not available, so $L$-shell parameterization is used and the |
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119 | following scaling relation |
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120 | \cite{hion.ICRU49,hion.40} is applied: |
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121 | \begin{equation} |
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122 | \label{hion.dd} |
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123 | C_{\nu} = V_{\nu}C_L(\theta_L,H_{\nu}\eta_L), \;\; V_{\nu}=\frac{n_{\nu}}{n_L}, \;\; H_{\nu}=\frac{J_{\nu}}{J_L}, |
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124 | \end{equation} |
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125 | where $V_{\nu}$ is a vertical scaling factor proportional to number of |
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126 | electrons at the shell $n_{\nu}$. |
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127 | The contribution of the shell correction term is about 10\% for protons |
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128 | at $T = 2 MeV$. |
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129 | |
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130 | \subsubsection{Density Correction} |
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131 | |
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132 | $\delta$ is a correction term which takes into account the reduction in energy |
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133 | loss due to the so-called {\it density effect}. This becomes important at |
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134 | high energies because media have a tendency to become polarized as the |
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135 | incident particle velocity increases. As a consequence, the atoms in a |
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136 | medium can no longer be considered as isolated. To correct for this effect |
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137 | the formulation of Sternheimer~\cite{hion.sternheimer} is used: |
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138 | \input{electromagnetic/utils/densityeffect} |
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139 | |
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140 | \subsubsection{High Order Corrections} |
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141 | |
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142 | High order corrections term to Bethe-Bloch formula (\ref{hion.d}) can |
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143 | be expressed as |
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144 | \begin{equation} |
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145 | \label{hion.cor} |
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146 | F = G - S + 2(z L_1 + z^2 L_2), |
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147 | \end{equation} |
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148 | where G is the Mott correction term, S is the finite size correction term, |
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149 | $L_1$ is the Barkas correction, $L_2$ is the Bloch correction. The Mott term |
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150 | \cite{hion.ahlen} describes the close-collision corrections tend to become |
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151 | more important at large velocities and higher charge of projectile. |
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152 | The Fermi result is used: |
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153 | \begin{equation} |
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154 | G = \pi\alpha z\beta. |
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155 | \end{equation} |
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156 | The Barkas correction term describes distant collisions. The parameterization |
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157 | of Ref. is expressed in the form: |
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158 | \begin{equation} |
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159 | L_1 = \frac{1.29 F_A(b/x^{1/2})}{Z^{1/2}x^{3/2}}, \;\; x = \frac{\beta^2}{Z\alpha^2}, |
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160 | \end{equation} |
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161 | where $F_A$ is tabulated function \cite{hion.Ashley}, b is scaled minimum impact parameter |
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162 | shown in Table \ref{hion.t1}. This and other corrections depending on atomic |
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163 | properties are assumed to be additive for mixtures and compounds. |
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164 | \begin{table}[hbt] |
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165 | \begin{centering} |
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166 | \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} |
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167 | \hline |
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168 | $Z$ & 1 ($H_2$ gas) & 1 & 2 & 3 - 10 & 11 - 17 & 18 & 19 - 25 & 26 - 50 & $>$ 50\\ \hline |
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169 | $d$ & 0.6 & 1.8 & 0.6 & 1.8 & 1.4 & 1.8 & 1.4 & 1.35 & 1.3\\ \hline |
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170 | \end{tabular} |
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171 | \caption{Scaled minimum impact parameter b \cite{hion.ICRU49}.} |
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172 | \label{hion.t1} |
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173 | \end{centering} |
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174 | \end{table} |
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175 | For the Bloch correction term the classical expression \cite{hion.ICRU49} is following: |
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176 | \begin{equation} |
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177 | z^2L_2 = -y^2 \sum^{\infty}_{n=1} \frac{1}{n(n^2 + y^2)}, \;\; y = \frac{z\alpha}{\beta}. |
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178 | \end{equation} |
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179 | The finite size correction term takes into account the space |
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180 | distribution of charge of |
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181 | the projectile particle. For muon it is zero, for hadrons this term become |
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182 | visible at energies above few hundred GeV and the following parameterization |
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183 | \cite{hion.ahlen} is used: |
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184 | \begin{equation} |
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185 | S = ln(1 + q), \;\; q = \frac{2 m_e T_{max}}{ \varepsilon^2}, |
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186 | \end{equation} |
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187 | where $T_{max}$ is given in relation (\ref{hion.c}), $\varepsilon$ is proportional to |
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188 | the inverse effective radius of the projectile (Table \ref{hion.t2}). |
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189 | \begin{table}[hbt] |
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190 | \begin{centering} |
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191 | \begin{tabular}{|c|c|} |
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192 | \hline |
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193 | mesons, spin = 0 ($\pi^{\pm}$, $K^{\pm}$) & $0.736\;GeV$\\ \hline |
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194 | baryons, spin = 1/2 & $0.843\;GeV$\\ \hline |
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195 | ions & $0.843\; A^{1/3}\;GeV$\\ \hline |
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196 | \end{tabular} |
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197 | \caption{The values of the $\varepsilon$ parameter for different particle types.} |
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198 | \label{hion.t2} |
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199 | \end{centering} |
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200 | \end{table} |
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201 | All these terms |
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202 | break scaling relation (\ref{enloss.sc}) if the projectile particle charge differs from |
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203 | $\pm$1. To take this circumstance into account in {\it G4ionIonisation} process |
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204 | at initialisation time |
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205 | the term $F$ is ignored for the computation of the $dE/dx$ table. |
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206 | At run time this term is taken into account by adding to the mean energy loss a value |
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207 | \begin{equation} |
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208 | \Delta T' = 2 \pi r_e^2 mc^2 n_{el} \frac{z^2}{\beta^2} F\Delta s, |
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209 | \end{equation} |
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210 | where $\Delta s$ is the {\it true step length} and $F$ is the high order correction |
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211 | term (\ref{hion.cor}). |
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212 | |
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213 | \subsubsection{Parameterizations at Low Energies} |
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214 | |
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215 | For scaled energies below $T_{lim} = 2\;MeV$ shell correction becomes very |
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216 | large and precision of the |
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217 | Bethe-Bloch formula degrades, so parameterisation of evaluated data |
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218 | for stopping powers at low energies is required. |
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219 | These parameterisations for all atoms |
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220 | is available from ICRU'49 report \cite{hion.ICRU49}. The proton parametrisation is used |
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221 | in {\it G4BraggModel}, which is included by default in the process {\it G4hIonisation}. |
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222 | The alpha particle parameterisation is used in the |
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223 | {\it G4BraggIonModel}, which is included by default in the process {\it G4ionIonisation}. |
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224 | To provide a smooth transition between low-energy and high-energy models the modified |
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225 | energy loss expression is used for high energy |
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226 | \begin{equation} |
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227 | S(T) = S_H (T) + (S_L(T_{lim}) - S_H(T_{lim}))\frac{T_{lim}}{T}, \;\; T > T_{lim}, |
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228 | \end{equation} |
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229 | where $S$ is smoothed stopping power, $S_H$ is stopping power from formula (\ref{hion.d}) |
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230 | and $S_L$ is the low-energy parameterisation. |
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231 | |
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232 | The precision of Bethe-Bloch formula for $T>10 MeV$ is within 2\%, below the precision |
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233 | degrades and at $1 keV$ only 20\% may be garanteed. In the energy interval $1 - 10 MeV$ |
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234 | the quality of description of the stopping power varied from atom to atom. To |
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235 | provide more stable and precise parameterisation the data from |
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236 | the NIST databases are included inside the standard package. These data are provided for |
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237 | 74 materials of the NIST material database \cite{hion.nist}. The data from the PSTAR database |
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238 | are included into {\it G4BraggModel}. The data from the ASTAR database |
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239 | are included into {\it G4BraggIonModel}. So, if Geant4 material is defined as a NIST |
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240 | material, than NIST data are used for low-energy parameterisation of stopping power. |
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241 | If material is not from the NIST database, then the ICRU'49 parameterisation is used. |
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242 | |
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243 | |
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244 | \subsubsection{Nuclear Stopping} |
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245 | |
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246 | Nuclear stopping due to elastic ion-ion scattering since Geant4 v9.3 |
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247 | can be simulated with the continuous process |
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248 | {\it G4NuclearStopping}. By default |
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249 | this correction is active and the ICRU'49 parameterisation \cite{hion.ICRU49} is used, |
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250 | which is implemented in the model class {\it G4ICRU49NuclearStoppingModel}. |
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251 | |
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252 | |
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253 | \subsection{Total Cross Section per Atom and Mean Free Path} |
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254 | |
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255 | For $T \gg I $ the differential cross section can be written as |
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256 | \begin{equation} |
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257 | \label{hion.i} |
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258 | \frac{d\sigma }{dT} = 2\pi r_e^2 mc^2 Z \frac{z_p^2}{\beta^2} \frac{1}{T^2} |
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259 | \left[ 1 - \beta^2 \frac{T}{T_{max}} + \frac{T^2}{2E^2} \right] |
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260 | \end{equation} |
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261 | \cite{hion.pdg}. In {\sc Geant4} $T_{cut} \geq 1$ keV. Integrating from |
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262 | $T_{cut}$ to $T_{max}$ gives the total cross section per atom : |
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263 | \begin{eqnarray} |
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264 | \label{hion.j} |
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265 | \sigma (Z,E,T_{cut}) & = & \frac {2\pi r_e^2 Z z_p^2}{\beta^2} mc^2 \times |
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266 | \\ & & \left[ \left( \frac{1}{T_{cut}} - \frac{1}{T_{max}} \right) |
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267 | - \frac{\beta^2}{T_{max}} \ln \frac{T_{max}}{T_{cut}} |
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268 | + \frac{T_{max} - T_{cut}}{2E^2} |
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269 | \right] \nonumber |
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270 | \end{eqnarray} |
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271 | The last term is for spin $1/2$ only. In a given material the mean free path |
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272 | is: |
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273 | \begin{equation} |
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274 | \begin{array}{lll} |
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275 | \lambda = (n_{at} \cdot \sigma)^{-1} & or & |
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276 | \lambda = \left( \sum_i n_{ati} \cdot \sigma_i \right)^{-1} |
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277 | \end{array} |
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278 | \end{equation} |
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279 | The mean free path is tabulated during initialization as a function of the |
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280 | material and of the energy for all kinds of charged particles. |
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281 | |
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282 | \subsection{Simulating Delta-ray Production} |
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283 | |
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284 | A short overview of the sampling method is given in Chapter \ref{secmessel}. |
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285 | Apart from the normalization, the cross section \ref{hion.i} can be |
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286 | factorized : |
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287 | \begin{eqnarray} |
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288 | \frac{d\sigma}{dT}=f(T) g(T) &with& T \in [T_{cut}, \ T_{max}] |
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289 | \end{eqnarray} |
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290 | where |
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291 | \begin{eqnarray} |
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292 | f(T) &=& \left(\frac{1}{T_{cut}} - \frac{1}{T_{max}} \right) \frac{1}{T^2} \\ |
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293 | g(T) &=& 1 - \beta^2 \frac{T}{T_{max}} + \frac{T^2}{2E^2} . |
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294 | \end{eqnarray} |
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295 | The last term in $g(T)$ is for spin $1/2$ only. The energy $T$ is chosen by |
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296 | \begin{enumerate} |
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297 | \item sampling $T$ from $f(T)$ |
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298 | \item calculating the rejection function $g(T)$ and accepting the sampled |
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299 | $T$ with a probability of $g(T)$. |
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300 | \end{enumerate} |
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301 | After the successful sampling of the energy, the direction |
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302 | of the scattered electron is generated with respect to the direction of the |
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303 | incident particle. The azimuthal angle $\phi$ is generated isotropically. |
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304 | The polar angle $\theta$ is calculated from energy-momentum conservation. |
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305 | This information is used to calculate the energy and momentum of both |
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306 | scattered particles and to transform them into the {\em global} coordinate |
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307 | system. |
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308 | |
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309 | \subsubsection{Ion Effective Charge} |
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310 | |
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311 | As ions penetrate matter they exchange electrons with the medium. In the |
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312 | implementation of {\it G4ionIonisation} the effective charge approach is |
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313 | used \cite{hion.Ziegler85}. |
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314 | A state of equilibrium between the ion and the medium is assumed, so that |
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315 | the ion's effective charge can be calculated as a function of its kinetic |
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316 | energy in a given material. Before and after each step the dynamic |
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317 | charge of the ion is recalculated and saved in $G4DynamicParticle$, where |
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318 | it can be used not only for energy loss calculations but also for the |
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319 | sampling of transportation in an electromagnetic field. |
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320 | |
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321 | The ion effective charge is expressed via |
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322 | the ion charge $z_i$ and the |
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323 | fractional effective charge of ion $\gamma_i$: |
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324 | \begin{equation} |
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325 | z_{eff} = \gamma_i z_i. |
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326 | \label{hlei.p} |
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327 | \end{equation} |
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328 | For helium ions |
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329 | fractional effective charge |
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330 | is parameterized for all elements |
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331 | \begin{eqnarray} |
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332 | (\gamma_{He})^2 & = &\left (1-\exp\left [-\sum_{j=0}^5{C_jQ^j}\right ]\right) |
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333 | \left ( 1 + \frac{ 7 + 0.05 Z }{1000} \exp( -(7.6-Q)^2 ) \right )^2, |
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334 | \nonumber \\ |
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335 | Q & = & \max ( 0, \ln T), |
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336 | \label{hion.q} |
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337 | \end{eqnarray} |
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338 | where the coefficients $C_j$ are the same for all elements, and the |
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339 | helium ion kinetic energy $T$ is in $keV/amu$. |
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340 | |
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341 | |
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342 | The following expression is used for heavy ions \cite{hion.BK}: |
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343 | \begin{equation} |
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344 | \gamma_i = \left ( q + \frac{1-q}{2} \left (\frac{v_0}{v_F} \right )^2 |
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345 | \ln {\left ( 1 + \Lambda^2 \right )} \right ) |
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346 | \left ( 1 + \frac{(0.18+0.0015Z)\exp(-(7.6-Q)^2)}{Z_i^2} \right ), |
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347 | \label{hion.s} |
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348 | \end{equation} |
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349 | where $q$ is |
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350 | the fractional average charge of the ion, |
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351 | $v_0$ is the Bohr velocity, |
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352 | $v_F$ is the Fermi velocity of |
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353 | the electrons in the target medium, and $\Lambda$ is |
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354 | the term taking into account the screening effect: |
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355 | \begin{equation} |
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356 | \Lambda = 10 \frac{v_F}{v_0} \frac{(1-q)^{2/3}}{Z_i^{1/3}(6+q)}. |
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357 | \label{hion.t} |
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358 | \end{equation} |
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359 | The Fermi velocity of the medium is of the same order as the Bohr velocity, and |
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360 | its exact value depends on the detailed electronic structure of the medium. |
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361 | The expression for the fractional average charge of the ion is the following: |
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362 | \begin{equation} |
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363 | q = [1 -\exp(0.803y^{0.3}-1.3167y^{0.6}-0.38157y-0.008983y^2)], |
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364 | \label{hion.u} |
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365 | \end{equation} |
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366 | where $y$ is a parameter that depends on the ion velocity $v_i$ |
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367 | \begin{equation} |
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368 | y = \frac{v_i}{v_0Z^{2/3}} \left ( 1 +\frac {v_F^2}{5v_i^2} \right ). |
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369 | \label{hion.v} |
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370 | \end{equation} |
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371 | The parametrisation of the effective charge of the ion applied |
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372 | if the kinetic energy is below limit value |
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373 | \begin{equation} |
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374 | T < 10 z_i \frac{M_i}{M_p}\;MeV, |
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375 | \label{hion.x} |
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376 | \end{equation} |
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377 | where $M_i$ is the ion mass and $M_p$ is the proton mass. |
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378 | |
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379 | |
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380 | \subsection{Status of this document} |
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381 | 09.10.98 created by L. Urb\'an. \\ |
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382 | 14.12.01 revised by M.Maire \\ |
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383 | 29.11.02 re-worded by D.H. Wright \\ |
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384 | 01.12.03 revised by V. Ivanchenko \\ |
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385 | 21.06.07 revised by V. Ivanchenko \\ |
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386 | |
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387 | \begin{latexonly} |
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388 | |
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389 | \begin{thebibliography}{99} |
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390 | |
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391 | \bibitem{hion.pdg} |
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392 | W.-M.~Yao et al., Jour. of Phys. G33 (2006) 1. |
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393 | \bibitem{hion.ahlen} |
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394 | S.P. Ahlen, Rev. Mod. Phys. 52 (1980) 121. |
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395 | \bibitem{hion.ICRU37} |
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396 | ICRU (A.~Allisy et al), |
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397 | Stopping Powers for Electrons and Positrons, |
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398 | ICRU Report 37, 1984. |
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399 | \bibitem{hion.ICRU49}ICRU (A.~Allisy et al), |
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400 | Stopping Powers and Ranges for Protons and Alpha |
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401 | Particles, |
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402 | ICRU Report 49, 1993. |
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403 | \bibitem{hion.37} |
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404 | M.C.~Walske, Phys. Rev. 88 (1952) 1283. |
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405 | \bibitem{hion.38} |
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406 | M.C.~Walske, Phys. Rev. 181 (1956) 940. |
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407 | \bibitem{hion.39} |
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408 | G.S.~Khandelwal, Nucl. Phys. A116 (1968) 97. |
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409 | \bibitem{hion.40} |
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410 | H.~Bichsel, Phys. Rev. A46 (1992) 5761. |
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411 | \bibitem{hion.sternheimer} |
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412 | R.M.~Sternheimer. Phys.Rev. B3 (1971) 3681. |
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413 | \bibitem{hion.Ashley} |
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414 | J.C.~Ashley, R.H.~Ritchie and W.~Brandt, Phys. Rev. A8 (1973) 2402. |
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415 | \bibitem{hion.nist} |
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416 | http://physics.nist.gov/PhysRevData/contents-radi.html |
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417 | \bibitem{hion.Ziegler85} |
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418 | J.F.~Ziegler, J.P.~Biersack, U.~Littmark, The Stopping |
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419 | and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985. |
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420 | \bibitem{hion.BK} |
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421 | W.~Brandt and M.~Kitagawa, Phys. Rev. B25 (1982) 5631. |
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422 | \end{thebibliography} |
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423 | |
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424 | \end{latexonly} |
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425 | |
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426 | \begin{htmlonly} |
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427 | |
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428 | \subsection{Bibliography} |
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429 | |
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430 | \begin{enumerate} |
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431 | \item{hion.pdg} |
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432 | W.-M.~Yao et al., Jour. of Phys. G33 (2006) 1. |
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433 | \item{hion.ICRU37} |
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434 | S.P. Ahlen, Rev. Mod. Phys. 52 (1980) 121. |
---|
435 | \item{hion.ICRU37} |
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436 | ICRU (A.~Allisy et al), |
---|
437 | Stopping Powers for Electrons and Positrons, |
---|
438 | ICRU Report 37, 1984. |
---|
439 | \item{hion.ICRU49}ICRU (A.~Allisy et al), |
---|
440 | Stopping Powers and Ranges for Protons and Alpha |
---|
441 | Particles, |
---|
442 | ICRU Report 49, 1993. |
---|
443 | \item{hion.37} |
---|
444 | M.C.~Walske, Phys. Rev. 88 (1952) 1283. |
---|
445 | \item{hion.38} |
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446 | M.C.~Walske, Phys. Rev. 181 (1956) 940. |
---|
447 | \item{hion.39} |
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448 | G.S.~Khandelwal, Nucl. Phys. A116 (1968) 97. |
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449 | \item{hion.40} |
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450 | H.~Bichsel, Phys. Rev. A46 (1992) 5761. |
---|
451 | \item{hion.sternheimer} |
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452 | R.M.~Sternheimer. Phys.Rev. B3 (1971) 3681. |
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453 | \item{hion.Ashley} |
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454 | J.C.~Ashley, R.H.~Ritchie and W.~Brandt, Phys. Rev. A8 (1973) 2402. |
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455 | \item{hion.nist} |
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456 | http://physics.nist.gov/PhysRevData/contents-radi.html |
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457 | \item{hion.Ziegler85} |
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458 | J.F.~Ziegler, J.P.~Biersack, U.~Littmark, The Stopping |
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459 | and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985. |
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460 | \item{hion.BK} |
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461 | W.~Brandt and M.~Kitagawa, Phys. Rev. B25 (1982) 5631. |
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462 | \end{enumerate} |
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463 | |
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464 | \end{htmlonly} |
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465 | |
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466 | |
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