1 | |
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2 | \section{Positron - Electron Pair Production by Muons} |
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3 | |
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4 | Direct electron pair production is one of the most important muon interaction |
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5 | processes. At TeV muon energies, the pair production cross section exceeds |
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6 | those of other muon interaction processes over a range of energy transfers |
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7 | between 100 MeV and 0.1$E_{\mu}$. The average energy loss for pair |
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8 | production increases linearly with muon energy, and in the TeV region this |
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9 | process contributes more than half the total energy loss rate. |
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10 | |
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11 | To adequately describe the number of pairs produced, the average energy loss |
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12 | and the stochastic energy loss distribution, the differential cross section |
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13 | behavior over an energy transfer range of |
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14 | 5~MeV~$\leq \epsilon \leq $~0.1~$\cdot E_{ \mu }$ must be accurately |
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15 | reproduced. This is is because the main contribution to the total cross |
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16 | section is given by transferred energies |
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17 | 5 MeV $\leq $ $\epsilon $ $\leq $ 0.01 $\cdot E_{ \mu }$, and because the |
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18 | contribution to the average muon energy loss is determined mostly in the |
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19 | region |
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20 | $0.001 \cdot E_{ \mu } \leq \epsilon \leq $ 0.1 $\cdot E_{\mu }$ . |
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21 | |
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22 | For a theoretical description of the cross section, the formulae of |
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23 | Ref.~\cite{pair.koko69} are used, along with a correction for finite nuclear |
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24 | size \cite{pair.koko71}. To take into account electron pair production in |
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25 | the field of atomic electrons, the inelastic atomic form factor contribution |
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26 | of Ref. \cite{pair.keln97} is also applied. |
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27 | |
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28 | |
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29 | \subsection{Differential Cross Section} |
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30 | |
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31 | \subsubsection{Definitions and Applicability} |
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32 | |
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33 | In the following discussion, these definitions are used: |
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34 | |
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35 | \begin{itemize} |
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36 | \item $m$ and $\mu$ are the electron and muon masses, respectively |
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37 | \item $E\equiv E_{\mu}$ is the total muon energy, $E = T +\mu $ |
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38 | \item $Z$ and $A$ are the atomic number and weight of the material |
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39 | \item $\epsilon$ is the total pair energy or, approximately, the muon energy |
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40 | loss $(E-E')$ |
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41 | \item $v = \epsilon/E$ |
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42 | \item $e = 2.718\dots$ |
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43 | \item $A^{\star} = 183$. |
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44 | \end{itemize} |
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45 | |
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46 | \noindent |
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47 | The formula for the differential cross section applies when: |
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48 | |
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49 | \begin{itemize} |
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50 | \item $E_{\mu} \gg \mu$ ($E \geq$ 2 -- 5 GeV) and |
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51 | $E_{\mu} \leq 10^{15}$ -- $10^{17} $ eV. If muon energies exceed this limit, |
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52 | the LPM (Landau Pomeranchuk Migdal) effect may become important, depending on |
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53 | the material |
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54 | |
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55 | \item the muon energy transfer $\epsilon$ lies between |
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56 | $\epsilon_{\rm min} = 4\,m$ and $\epsilon_{\rm max} = E_{\mu}- \frac{3 \sqrt{e}}{4}\,\mu \,Z^{1/3}$, although the formal lower limit is |
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57 | $\epsilon\gg 2\, m$, and the formal upper limit requires $E'_{\mu }\gg\mu$. |
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58 | |
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59 | \item $Z \leq$ 40 -- 50. For higher $Z$, the Coulomb correction is important |
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60 | but has not been sufficiently studied theoretically. |
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61 | \end{itemize} |
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62 | |
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63 | \subsubsection{Formulae} |
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64 | |
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65 | The differential cross section for electron pair production by muons |
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66 | $\sigma(Z,A,E,\epsilon)$ can be written as : |
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67 | \begin{equation} |
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68 | \label{mupair.a} |
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69 | \sigma(Z,A,E,\epsilon)=\frac{4}{3\pi}\,\frac{Z(Z+\zeta )} |
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70 | {A}\, N_{A}\,(\alpha r_{0})^{2}\, \frac{1-v}{\epsilon} |
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71 | \int_{0}^{\rho_{\rm max}}G(Z,E,v,\rho)\,d\rho , |
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72 | \end{equation} |
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73 | |
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74 | \noindent |
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75 | where |
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76 | $$G(Z,E,v,\rho ) = \Phi_e + (m/\mu)^2 \Phi_{\mu} , $$ |
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77 | $$\Phi_{e,\mu} = B_{e,\mu} L'_{e,\mu} $$ and |
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78 | $$\Phi_{e,\mu} = 0 \quad {\rm whenever} \quad \Phi_{e,\mu} < 0 . $$ |
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79 | |
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80 | \noindent |
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81 | $B_{e}$ and $B_{\mu}$ do not depend on $Z,A$, and are given by |
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82 | |
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83 | $$B_{e}=[(2+\rho^{2})(1+\beta)+\xi(3+\rho^{2})] |
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84 | \ln\left(1+\frac{1}{\xi}\right)+\frac{1-\rho^{2}-\beta}{1+\xi}-(3+\rho^{2}) ; $$ |
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85 | $$B_{e}\approx |
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86 | \frac{1}{2\xi}\,[(3-\rho^{2})+2\beta(1+\rho^{2})]\quad {\rm |
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87 | for}\quad\xi\geq 10^{3};$$ |
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88 | |
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89 | $$B_{\mu}=\left[(1+\rho^{2}) |
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90 | \left(1+\frac{3\beta}{2} |
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91 | \right)-\frac{1}{\xi}(1+2\beta) |
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92 | (1-\rho^{2})\right]\ln(1+\xi ) |
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93 | $$ |
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94 | $$ |
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95 | +\frac{\xi(1-\rho^{2}-\beta)}{1+\xi}+ |
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96 | (1+2\beta)(1-\rho^{2});$$ |
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97 | $$B_{\mu}\approx\frac{\xi}{2}\,[(5-\rho^{2})+\beta(3+\rho^{2})]\quad{\rm |
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98 | for}\quad\xi\leq10^{-3} |
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99 | ;$$ |
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100 | |
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101 | \noindent |
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102 | Also, |
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103 | % $B_{e},\:B_{\mu}$ do not depend on $\,Z,\:A$. |
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104 | |
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105 | $$\xi=\frac{\mu^{2} v^{2}}{4m^{2}}\, \frac{(1-\rho^{2})}{(1-v)};\quad |
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106 | \beta=\frac{v^{2}}{2(1-v)};$$ |
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107 | |
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108 | $$L'_{e}=\ln\frac{A^{*}Z^{-1/3}\sqrt{(1+\xi)(1+Y_{e})}} |
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109 | {1+\displaystyle\frac{2m\sqrt{e}A^{*}Z^{-1/3}(1+\xi)(1+Y_{e})}{Ev(1-\rho^{2})}} |
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110 | $$ |
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111 | $$ |
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112 | -\frac{1}{2}\ln\left[1+\left(\frac{3mZ^{1/3}}{2\mu}\right)^{2}(1+\xi)(1+Y_{e})\right]; |
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113 | $$ |
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114 | |
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115 | $$L'_{\mu}=\ln\frac{(\mu/m)A^{*}Z^{-1/3}\sqrt{(1+1/\xi)(1+Y_{\mu})}} |
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116 | {1+\displaystyle\frac{2m\sqrt{e}A^{*}Z^{-1/3}(1+\xi)(1+Y_{\mu})}{Ev(1-\rho^{2})}} |
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117 | $$ |
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118 | $$ |
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119 | -\ln\left[\frac{3}{2}\,Z^{1/3}\sqrt{(1+1/\xi)(1+Y_{\mu})}\right]. |
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120 | $$ |
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121 | % |
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122 | For faster computing, the expressions for $L'_{e,\mu }$ are |
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123 | further algebraically transformed. The functions $L'_{e,\mu }$ include the |
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124 | nuclear size correction \cite{pair.koko71} in comparison with parameterization |
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125 | \cite{pair.koko69} : |
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126 | % |
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127 | $$Y_{e}=\frac{5-\rho^{2}+4\,\beta\,(1+\rho^{2})} |
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128 | {2(1+3\beta)\ln(3+{1}/{\xi})-\rho^{2}-2\beta(2-\rho^{2})} ;$$ |
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129 | |
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130 | $$Y_{\mu}=\frac{4+\rho^{2}+3\,\beta\,(1+\rho^{2})} |
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131 | {(1+\rho^{2})(\frac{3}{2}+2\beta)\ln(3+\xi)+1-\frac{3}{2}\,\rho^{2}};$$ |
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132 | %$Y_{e,\mu}$~--~ approximations (\cite{pair.koko69}); there is |
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133 | %a possibility that they will be improved , as well as corrections (second |
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134 | % terms) in $L_{e,\mu}$. |
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135 | % |
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136 | $$\rho_{\rm max}=[1-6\mu^{2}/E^{2}(1-v)]\sqrt{1-4m/Ev}.$$ |
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137 | |
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138 | \subsubsection{Comment on the Calculation of the Integral $\int\!d\rho$ in |
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139 | Eq.~\ref{mupair.a}} |
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140 | |
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141 | The integral $\int\limits_{0}^{\rho_{\max}}G(Z,E,v,\rho)~d\rho$ is computed |
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142 | with the substitutions: |
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143 | \begin{eqnarray*} |
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144 | t&=& \ln(1-\rho),\\ |
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145 | 1 - \rho &= &\exp(t),\\ |
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146 | 1 + \rho& =& 2 - \exp(t),\\ |
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147 | 1 - \rho^{2} &=&e^{t}\,(2-e^{t}). |
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148 | \end{eqnarray*} |
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149 | |
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150 | \noindent |
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151 | After that, |
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152 | \begin{equation} |
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153 | \label{mupair.b} |
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154 | \int_{0}^{\rho_{\rm max}}G(Z,E,v,\rho)~d\rho = |
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155 | \int_{t_{\rm min}}^{0}G(Z,E,v,\rho )\,e^{t}\,dt , |
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156 | \end{equation} |
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157 | |
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158 | \noindent |
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159 | where |
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160 | |
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161 | $$t_{\rm |
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162 | min}=\ln\frac{\displaystyle\frac{4m}{\epsilon}+\frac{12\mu^{2}}{EE'}\left(1-\frac{4m} |
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163 | {\epsilon}\right)} |
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164 | {\displaystyle 1+\left(1-\frac{6\mu^{2}}{EE'}\right)\sqrt{1-\frac{4m}{\epsilon}}}. |
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165 | $$ |
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166 | |
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167 | To compute the integral of Eq.~\ref{mupair.b} with an accuracy better than |
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168 | 0.5\%, Gaussian quadrature with $N=8$ points is sufficient. |
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169 | |
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170 | The function $\zeta(E,Z)$ in Eq.~\ref{mupair.a} serves to take into account |
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171 | the process on atomic electrons (inelastic atomic form factor contribution). |
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172 | To treat the energy loss balance correctly, the following approximation, |
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173 | which is an algebraic transformation of the expression in |
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174 | Ref.~\cite{pair.keln97}, is used: |
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175 | |
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176 | $$\zeta(E,Z)=\frac{\displaystyle 0.073\ln\frac{E/\mu}{ |
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177 | 1+\gamma_{1}Z^{2/3}E/\mu} |
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178 | -0.26} |
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179 | {\displaystyle 0.058\ln\frac{E/\mu}{1+\gamma_{2}Z^{1/3}E/\mu} |
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180 | -0.14}; $$ |
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181 | $$\zeta(E,Z)=0\quad\mbox{if the numerator is negative.}$$ |
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182 | |
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183 | \noindent |
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184 | For E $\leq 35\,\mu, ~ \zeta(E,Z)=0$. Also |
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185 | $\gamma_{1}= 1.95\cdot 10^{-5} $ and $\gamma_{2}= 5.30\cdot 10^{-5} $. |
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186 | |
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187 | The above formulae make use of the Thomas-Fermi model which is not good |
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188 | enough for light elements. For hydrogen ($Z = 1$) the following parameters |
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189 | must be changed: \\ |
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190 | $A^{*}=183 ~\Rightarrow ~202.4;$\\ |
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191 | $\gamma_{1}= 1.95\cdot 10^{-5} ~\Rightarrow ~ 4.4\cdot 10^{-5};$\\ |
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192 | $\gamma_{2}= 5.30\cdot 10^{-5} ~\Rightarrow ~4.8\cdot 10^{-5}.$\\ |
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193 | |
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194 | \subsection{Total Cross Section and Restricted Energy Loss} |
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195 | |
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196 | If the user's cut for the energy transfer $\epsilon_{\rm cut}$ is |
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197 | greater than $\epsilon_{\min}$, the process is represented by |
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198 | continuous restricted energy loss for interactions with |
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199 | $\epsilon\le\epsilon_{\rm cut}$, and discrete collisions with |
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200 | $\epsilon>\epsilon_{\rm cut}$. Respective values of the total cross |
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201 | section and restricted energy loss rate are defined as: |
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202 | $$ |
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203 | \sigma_{\rm tot}=\int_{\epsilon_{\rm cut}}^{\epsilon_{\max}} |
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204 | \sigma(E,\epsilon)\,d\epsilon;\quad |
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205 | (dE/dx)_{\rm restr}=\int_{\epsilon_{\min}}^{\epsilon_{\rm cut}} |
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206 | \epsilon\,\sigma(E,\epsilon)\,d\epsilon. |
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207 | $$ |
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208 | For faster computing, $\ln\epsilon$ substitution and Gaussian quadratures are |
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209 | used. |
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210 | |
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211 | \subsection{Sampling of Positron - Electron Pair Production} |
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212 | |
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213 | The e$^+$e$^-$ pair energy $\epsilon_P$, is found numerically by solving the |
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214 | equation |
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215 | \begin{equation} |
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216 | \label{mupair.c} |
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217 | P = \int_{\epsilon_P}^{\epsilon_{\rm max}} \sigma (Z,A,T,\epsilon) d\epsilon |
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218 | \quad / \int_{cut}^{\epsilon_{\rm max}} \sigma (Z,A,T,\epsilon) d\epsilon |
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219 | \end{equation} |
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220 | or |
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221 | \begin{equation} |
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222 | \label{mupair.d} |
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223 | 1 - P = \int_{cut}^{\epsilon_P} \sigma (Z,A,T,\epsilon) d\epsilon |
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224 | \quad / \int_{cut}^{\epsilon_{\rm max}} \sigma (Z,A,T,\epsilon) d\epsilon |
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225 | \end{equation} |
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226 | |
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227 | To reach high sampling speed, solutions of Eqs.~\ref{mupair.c}, \ref{mupair.d} |
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228 | are tabulated at initialization time. Two 3-dimensional tables (referred to |
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229 | here as A and B) of $\epsilon_{P}(P,T,Z)$ are created, and then interpolation |
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230 | is used to sample $\epsilon_P$. |
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231 | |
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232 | \noindent |
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233 | The number and spacing of entries in the table are chosen as follows: |
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234 | \begin{itemize} |
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235 | \item a constant increment in $\ln T$ is chosen such that there are four |
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236 | points per decade in the range $T_{\rm min}- T_{\rm max}$. The default range |
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237 | of muon kinetic energies in Geant4 is $T=1\:{\rm GeV}-1000\:{\rm PeV}$. |
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238 | |
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239 | \item a constant increment in $\ln Z$ is chosen. The shape of the sampling |
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240 | distribution does depend on $Z$, but very weakly, so that eight points in the |
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241 | range $1\leq Z\leq 128$ are sufficient. There is practically no dependence |
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242 | on the atomic weight $A$. |
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243 | |
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244 | \item for probabilities $P \leq 0.5$, Eq.~\ref{mupair.c} is used and Table~A |
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245 | is computed with a constant increment in $\ln P$ in the range |
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246 | $10^{-7}\leq P \leq 0.5$. The number of points in $\ln P$ for Table~A is |
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247 | about 100. |
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248 | |
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249 | \item for $P \geq 0.5$, Eq.~\ref{mupair.d} is used and Table~B is computed |
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250 | with a constant increment in $\ln(1-P)$ in the range |
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251 | $10^{-5} \leq (1-P) \leq 0.5$. In this case 50 points are sufficient. |
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252 | \end{itemize} |
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253 | |
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254 | \noindent |
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255 | The values of $\ln (\epsilon_{P}-cut$) are stored in both Table~A and Table~B. |
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256 | |
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257 | |
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258 | To create the ``probability tables" for each $(T, Z)$ pair, the following |
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259 | procedure is used: |
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260 | |
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261 | \begin{itemize} |
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262 | \item a temporary table of $\sim$ 2000 values of |
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263 | $\epsilon \cdot \sigma(Z,A,T,\epsilon)$ is constructed with a constant |
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264 | increment ($\sim$ 0.02) in $\ln \epsilon$ in the range |
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265 | $(cut, \epsilon_{\max})$. $\epsilon$ is taken in the middle of the |
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266 | corresponding bin in $\ln \epsilon$. |
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267 | |
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268 | \item the accumulated cross sections |
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269 | $$\sigma_{1}=\int_{\ln\epsilon}^{\ln\epsilon_{\max}} |
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270 | \epsilon \, \sigma (Z,A,T,\epsilon)\, d (\ln\epsilon) $$ |
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271 | and |
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272 | $$\sigma_{2}=\int_{\ln(cut)}^{\ln\epsilon} |
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273 | \epsilon \, \sigma (Z,A,T,\epsilon)\, d (\ln\epsilon ) $$ |
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274 | |
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275 | are calculated by summing the temporary table over the values above |
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276 | $\ln \epsilon$ (for $\sigma_1$) and below $\ln \epsilon$ (for $\sigma_2$) |
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277 | and then normalizing to obtain the accumulated probability functions. |
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278 | |
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279 | \item finally, values of $\ln(\epsilon_{P} - cut $) for corresponding values |
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280 | of $\ln P$ and $\ln (1-P)$ are calculated by linear interpolation of the |
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281 | above accumulated probabilities to form Tables A and B. The monotonic |
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282 | behavior of the accumulated cross sections is very useful in speeding up |
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283 | the interpolation procedure. |
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284 | |
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285 | \end{itemize} |
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286 | |
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287 | The random transferred energy corresponding to a probability $P$, is then |
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288 | found by linear interpolation in $\ln Z$ and $\ln T$, and a cubic |
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289 | interpolation in $\ln P$ for Table A or in $\ln (1-P)$ for Table B. |
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290 | For $P \leq 10^{-7}$ and $(1-P) \leq 10^{-5}$, linear extrapolation using |
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291 | the entries at the edges of the tables may be safely used. Electron pair |
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292 | energy is related to the auxiliary variable |
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293 | $x = \ln (\epsilon_{P} - cut)$ found by the trivial interpolation |
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294 | $\epsilon_{P} = e^{x} + cut$. |
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295 | |
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296 | Similar to muon bremsstrahlung (section \ref{secmubrem}), this sampling |
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297 | algorithm does not re-initialize the tables for user cuts greater than |
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298 | $cut_{min}$. Instead, the probability variable is redefined as |
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299 | $$ P' = P \sigma_{\rm tot}(cut_{user}) / \sigma_{\rm tot}(cut_{min}),$$ |
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300 | and $P'$ is used for sampling. |
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301 | |
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302 | In the simulation of the final state, the muon deflection angle (which is |
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303 | of the order of $m/E$) is neglected. The procedure for sampling the energy |
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304 | partition between $e^+$ and $e^-$ and their emission angles is similar to |
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305 | that used for the $\gamma \to e^+\,e^-$ conversion. |
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306 | |
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307 | \subsection{Status of this document} |
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308 | |
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309 | 12.10.98 created by R.Kokoulin and A.Rybin\\ |
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310 | 18.05.00 edited by S.Kelner, R.Kokoulin, and A.Rybin\\ |
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311 | 27.01.03 re-written by D.H. Wright |
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312 | |
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313 | \begin{latexonly} |
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314 | |
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315 | \begin{thebibliography}{99} |
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316 | |
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317 | \bibitem{pair.koko69} R.P.Kokoulin and A.A.Petrukhin, |
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318 | Proc. 11th Intern. Conf. on Cosmic Rays, Budapest, 1969 |
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319 | [Acta Phys. Acad. Sci. Hung.,{\bf 29, Suppl.4}, |
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320 | p.277, 1970]. |
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321 | |
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322 | \bibitem{pair.koko71} R.P.Kokoulin and A.A.Petrukhin, |
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323 | Proc. 12th Int. Conf. on Cosmic Rays, Hobart, 1971, |
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324 | {\bf vol.6}, p.2436. |
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325 | |
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326 | \bibitem{pair.keln97} S.R.Kelner, Phys. Atomic Nuclei, |
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327 | {\bf 61} (1998) 448. |
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328 | |
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329 | \end{thebibliography} |
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330 | |
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331 | \end{latexonly} |
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332 | |
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333 | \begin{htmlonly} |
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334 | |
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335 | \subsection{Bibliography} |
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336 | |
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337 | \begin{enumerate} |
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338 | \item R.P.Kokoulin and A.A.Petrukhin, |
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339 | Proc. 11th Intern. Conf. on Cosmic Rays, Budapest, 1969 |
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340 | [Acta Phys. Acad. Sci. Hung.,{\bf 29, Suppl.4}, p.277, 1970]. |
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341 | |
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342 | \item R.P.Kokoulin and A.A.Petrukhin, |
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343 | Proc. 12th Int. Conf. on Cosmic Rays, Hobart, 1971, {\bf vol.6}, p.2436. |
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344 | |
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345 | \item S.R.Kelner, Phys. Atomic Nuclei, {\bf 61} (1998) 448. |
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346 | \end{enumerate} |
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347 | |
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348 | \end{htmlonly} |
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