1 | |
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2 | \section[Bremsstrahlung]{Bremsstrahlung} \label{secmubrem} |
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3 | |
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4 | Bremsstrahlung dominates other muon interaction processes in the region of |
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5 | catastrophic collisions ($v \geq 0.1$ ), that is at "moderate" muon |
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6 | energies above the kinematic limit for knock--on electron production. |
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7 | At high energies ($E \geq 1$ TeV) this process contributes about |
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8 | 40\% of the average muon energy loss. |
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9 | |
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10 | \subsection{Differential Cross Section} |
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11 | |
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12 | The differential cross section for muon bremsstrahlung (in units of |
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13 | ${\rm cm}^{2}/(\mbox{g GeV})$) |
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14 | can be written as |
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15 | \begin{eqnarray} |
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16 | \label{mubrem.a} |
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17 | \frac{d \sigma (E,\epsilon,Z,A)}{d \epsilon}&=& |
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18 | \frac{16}{3} \alpha N_A (\frac{m}{\mu} r_{e})^2 |
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19 | \frac{1}{\epsilon A} Z(Z \Phi_n + \Phi_e)(1-v+\frac{3}{4} v^2)\nonumber\\ |
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20 | &=&0 \quad {\rm if} \quad \epsilon \geq \epsilon_{\rm max} = E-\mu , |
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21 | \end{eqnarray} |
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22 | where $\mu$ and $m$ are the muon and electron masses, $Z$ and $A$ are the |
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23 | atomic number and atomic weight of the material, and $N_{A}$ is Avogadro's |
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24 | number. If $E$ and $T$ are the initial total and kinetic energy of the |
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25 | muon, and $\epsilon$ is the emitted photon energy, then |
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26 | $\epsilon = E - E'$ and the relative energy transfer $v = \epsilon /E$. |
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27 | |
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28 | $\Phi_{n}$ represents the contribution of the nucleus and can be expressed |
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29 | as |
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30 | \begin{eqnarray*} |
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31 | \Phi_{n} &=& \ln \frac {BZ^{-1/3}(\mu + \delta (D_{n}' \sqrt{e} -2))} |
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32 | {D_{n}'(m+ \delta \sqrt{e}BZ^{-1/3})} ; \\ |
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33 | &=& 0 \quad {\rm if} \quad {\rm negative}. |
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34 | \end{eqnarray*} |
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35 | $\Phi_{e}$ represents the contribution of the electrons and can be expressed |
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36 | as |
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37 | \begin{eqnarray*} |
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38 | \Phi_{e} &=& \ln \frac {B'Z^{-2/3} \mu } |
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39 | {\left(1+ \displaystyle\frac{\delta \mu}{m^{2} \sqrt{e}}\right)(m+ \delta |
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40 | \sqrt{e} |
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41 | B'Z^{-2/3}) |
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42 | }; \\ |
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43 | &=& 0 \quad {\rm if} \quad \epsilon \geq \epsilon'_{\rm max} = E/(1+ |
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44 | \mu^{2}/2mE); \\ |
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45 | &=& 0 \quad {\rm if} \quad {\rm negative}. |
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46 | \end{eqnarray*} |
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47 | In $\Phi_n$ and $\Phi_e$, for all nuclei except hydrogen, |
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48 | \begin{eqnarray*} |
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49 | \delta &=& \mu^{2} \epsilon /2EE' = \mu^{2} v/2(E- \epsilon);\\ |
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50 | D'_{n} &=& D_{n}^{(1-1/Z)}, \quad D_{n}= 1.54A^{0.27}; \\ |
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51 | B &=& 183, \quad B'=1429, \quad \sqrt{e}=1.648(721271). |
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52 | \end{eqnarray*} |
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53 | % |
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54 | For hydrogen ($Z$=1) $B = 202.4,\: B' = 446, \: D_{n}' = D_{n}$. |
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55 | |
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56 | |
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57 | These formulae are taken mostly from Refs. \cite{brem.kel95} and |
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58 | \cite{brem.kel97}. They include improved nuclear size corrections in |
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59 | comparison with Ref. \cite{brem.petr68} in the region $v \sim 1$ and low $Z$. |
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60 | Bremsstrahlung on atomic electrons (taking into account target recoil and |
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61 | atomic binding) is introduced instead of a rough substitution $Z(Z+1)$. |
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62 | A correction for processes with nucleus excitation is also |
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63 | included \cite{brem.andr94}. |
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64 | |
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65 | \subsubsection{Applicability and Restrictions of the Method} |
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66 | |
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67 | The above formulae assume that: \\ |
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68 | 1. $E \gg \mu $, hence the ultrarelativistic approximation is used; \\ |
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69 | 2. $E \leq 10^{20}$ eV; above this energy, LPM suppression can be expected;\\ |
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70 | 3. $v \geq 10^{-6}$ ; below $10^{-6}$ Ter-Mikaelyan suppression takes place. |
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71 | However, in the latter region the cross section of muon bremsstrahlung is |
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72 | several orders of magnitude less than that of other processes.\\ |
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73 | The Coulomb correction (for high $Z$) is not included. However, existing |
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74 | calculations \cite{brem.andr97} show that for muon bremsstrahlung this |
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75 | correction is small. |
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76 | |
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77 | \subsection{Continuous Energy Loss} |
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78 | |
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79 | The restricted energy loss for muon bremsstrahlung $(d E/ dx)_{\rm rest}$ |
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80 | with relative transfers $v = \epsilon / (T+ \mu) \leq v_{\rm cut}$ |
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81 | can be calculated as follows : |
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82 | $$ |
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83 | \left(\frac{d E}{d x}\right)_{\rm rest} |
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84 | = \int_{0}^{\epsilon_{\rm cut}} \epsilon\,\sigma (E,\epsilon )\,d\epsilon |
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85 | = |
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86 | (T+\mu ) \int_{0}^{v_{\rm cut}}\epsilon\,\sigma (E,\epsilon)\,dv\,. |
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87 | $$ |
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88 | % |
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89 | If the user cut $v_{\rm cut} \geq v_{\rm max}=T/(T+ \mu)$, the total |
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90 | average energy loss is calculated. Integration is done using Gaussian |
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91 | quadratures, and binning provides an accuracy better than about 0.03\% for |
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92 | $T = 1$ GeV, $Z=1$. This rapidly improves with increasing $T$ and $Z$. |
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93 | |
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94 | |
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95 | \subsection{Total Cross Section} |
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96 | |
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97 | The integration of the differential cross section over $d\epsilon$ gives the |
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98 | total cross section for muon bremsstrahlung: |
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99 | \begin{equation} |
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100 | \label{mubrem.b} |
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101 | \sigma_{\rm tot} (E,\epsilon_{\rm cut}) |
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102 | = \int_{\epsilon_{\rm cut}}^{\epsilon_{\rm max}}\sigma (E,\epsilon ) |
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103 | d \epsilon = |
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104 | \int_{\ln v_{\rm cut}}^{\ln v_{\rm max}}\epsilon \sigma (E,\epsilon) |
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105 | d(\ln v) , |
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106 | \end{equation} |
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107 | where $v_{\rm max}=T/(T+ \mu)$. If $v_{\rm cut} \geq v_{\rm max}$ , |
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108 | $\sigma_{\rm tot}=0$. |
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109 | |
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110 | \subsection{Sampling} |
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111 | |
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112 | The photon energy $\epsilon_{p}$ is found by numerically solving the |
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113 | equation : |
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114 | $$ P \:= \int_{\epsilon_{p}}^{\epsilon_{\rm max}} \sigma (E,\epsilon,Z,A |
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115 | ) \, d \epsilon |
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116 | \left/ \int_{\epsilon_{\rm cut}}^{\epsilon_{\rm max}} |
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117 | \sigma (E,\epsilon,Z,A ) \, d \epsilon\right. . |
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118 | $$ |
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119 | Here $P$ is the random uniform probability, $\epsilon_{\rm max}=T$, and |
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120 | $\epsilon_{\rm cut}=(T+\mu) \cdot v_{\rm cut}$. $v_{min.cut}=10^{-5}$ is |
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121 | the minimal relative energy transfer adopted in the algorithm. |
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122 | |
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123 | For fast sampling, the solution of the above equation is tabulated at |
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124 | initialization time for selected $Z$, $T$ and $P$. During simulation, this |
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125 | table is interpolated in order to find the value of $\epsilon_{p}$ |
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126 | corresponding to the probability $P$. |
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127 | |
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128 | The tabulation routine uses accurate functions for the differential cross |
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129 | section. The table contains values of |
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130 | \begin{equation} |
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131 | \label{mubrem.c} |
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132 | x_p = \ln (v_p / v_{\rm max})/\ln (v_{\rm max}/v_{\rm cut}) , |
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133 | \end{equation} |
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134 | where $v_{p} = \epsilon_{p}/(T+ \mu)$ and $v_{\rm max} = T/(T+ \mu)$. |
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135 | Tabulation is performed in the range |
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136 | $1 \leq Z \leq 128$, $1 \leq T \leq 1000$~PeV, $10^{-5} \leq P \leq 1$ |
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137 | with constant logarithmic steps. Atomic weight (which is a required |
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138 | parameter in the cross section) is estimated here with an iterative solution of the approximate relation: |
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139 | $$ A = Z\,(2+0.015\,A^{2/3}).$$ |
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140 | For $Z=1$, $A=1$ is used. |
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141 | |
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142 | To find $x_{p}$ (and thus $\epsilon_{p}$) corresponding to a given |
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143 | probability $P$, the sampling method performs a linear interpolation in |
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144 | $\ln Z$ and $\ln T$, and a cubic, 4 point Lagrangian interpolation |
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145 | in $\ln P$. For $P \leq P_{\rm min}$, a linear interpolation in $(P,x)$ |
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146 | coordinates is used, with $x = 0$ at $P = 0$. Then the energy |
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147 | $\epsilon_p$ is obtained from the inverse transformation of \ref{mubrem.c} : |
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148 | % |
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149 | $$\epsilon_{p} = (T+ \mu ) v_{\rm max} (v_{\rm max}/v_{\rm cut})^{x_{p}} $$ |
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150 | % |
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151 | The algorithm with the parameters described above has been |
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152 | tested for various $Z$ and $T$. It reproduces the differential cross section |
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153 | to within 0.2 -- 0.7 \% for $T \geq 10$~GeV. The average total energy loss |
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154 | is accurate to within 0.5\%. While accuracy improves with increasing $T$, |
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155 | satisfactory results are also obtained for $1 \leq T \leq 10$~GeV. |
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156 | |
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157 | It is important to note that this sampling scheme allows the generation of |
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158 | $\epsilon_{p}$ for different user cuts on $v$ which are above |
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159 | $v_{\rm min.cut}$. To perform such a simulation, it is sufficient to |
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160 | define a new probability variable |
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161 | % |
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162 | $$P' = P \: \sigma_{\rm tot} \: (v_{\rm user.cut}) / \sigma_{\rm tot} (v_{\rm |
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163 | min.cut})$$ |
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164 | % |
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165 | and use it in the sampling method. Time consuming re-calculation of the |
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166 | 3-dimensional table is therefore not required because only the tabulation |
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167 | of $\sigma_{\rm tot}(v_{\rm user.cut})$ is needed. |
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168 | |
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169 | The small-angle, ultrarelativistic approximation is used for the simulation |
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170 | (with about 20\% accuracy at $\theta\le\theta^*\approx1$) of the angular |
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171 | distribution of the final state muon and photon. Since the target recoil is |
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172 | small, the muon and photon are directed symmetrically (with equal transverse momenta and coplanar with the initial muon): |
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173 | \begin{equation} |
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174 | p_{\perp \mu} = p_{\perp \gamma}, \quad {\rm where} \quad p_{\perp \mu}= |
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175 | E' \theta_{\mu}, \quad p_{\perp \gamma} = \epsilon \theta_{\gamma} . |
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176 | \end{equation} |
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177 | $\theta_{\mu}$ and $\theta_{\gamma}$ are muon and photon emission angles. |
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178 | The distribution in the variable |
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179 | $r=E\theta_{\gamma}/\mu$ is given by |
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180 | \begin{equation} |
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181 | f(r) dr \sim r dr/(1+r^2)^2 . |
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182 | \end{equation} |
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183 | Random angles are sampled as follows: |
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184 | \begin{equation} |
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185 | \theta_{\gamma} = \frac{\mu}{E} r \quad |
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186 | \theta_{\mu} = \frac{\epsilon}{E'} \theta_{\gamma} , |
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187 | \end{equation} |
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188 | where |
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189 | $$ |
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190 | r=\sqrt{\frac{a}{1-a}}\,,\quad a=\xi\,\frac{r_{\rm \max}^2}{1+r_{\rm max}^2}\,,\quad |
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191 | r_{\max}=\min(1,E'/\epsilon)\cdot E\,\theta^*/\mu\,,$$ |
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192 | and $\xi$ is a random number uniformly distributed between 0 and 1. |
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193 | |
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194 | \subsection{Status of this document} |
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195 | |
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196 | 09.10.98 created by R.Kokoulin and A.Rybin\\ |
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197 | 17.05.00 updated by S.Kelner, R.Kokoulin and A.Rybin\\ |
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198 | 30.11.02 re-written by D.H. Wright\\ |
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199 | |
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200 | \begin{latexonly} |
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201 | |
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202 | \begin{thebibliography}{599} |
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203 | |
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204 | \bibitem{brem.kel95} |
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205 | S.R.Kelner, R.P.Kokoulin, A.A.Petrukhin. Preprint MEPhI 024-95, Moscow, 1995; |
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206 | CERN SCAN-9510048. |
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207 | \bibitem{brem.kel97} |
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208 | S.R.Kelner, R.P.Kokoulin, A.A.Petrukhin. Phys. Atomic Nuclei, {\bf 60} (1997) 576. |
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209 | \bibitem{brem.petr68} |
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210 | A.A.Petrukhin, V.V.Shestakov. Canad.J.Phys., {\bf 46} (1968) S377. |
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211 | \bibitem{brem.andr94} |
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212 | Yu.M.Andreyev, L.B.Bezrukov, E.V.Bugaev. Phys. Atomic Nuclei, {\bf 57} (1994) |
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213 | 2066. |
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214 | \bibitem{brem.andr97} |
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215 | Yu.M.Andreev, E.V.Bugaev, Phys. Rev. D, {\bf 55} (1997) 1233. |
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216 | \end{thebibliography} |
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217 | |
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218 | \end{latexonly} |
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219 | |
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220 | \begin{htmlonly} |
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221 | |
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222 | \subsection{Bibliography} |
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223 | |
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224 | \begin{enumerate} |
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225 | \item S.R.Kelner, R.P.Kokoulin, A.A.Petrukhin. Preprint MEPhI 024-95, Moscow, |
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226 | 1995; CERN SCAN-9510048. |
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227 | \item S.R.Kelner, R.P.Kokoulin, A.A.Petrukhin. Phys. Atomic Nuclei, {\bf 60} |
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228 | (1997) 576. |
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229 | \item A.A.Petrukhin, V.V.Shestakov. Canad.J.Phys., {\bf 46} (1968) S377. |
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230 | \item Yu.M.Andreyev, L.B.Bezrukov, E.V.Bugaev. Phys. Atomic Nuclei, {\bf 57} |
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231 | (1994) 2066. |
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232 | \item Yu.M.Andreev, E.V.Bugaev, Phys. Rev. D, {\bf 55} (1997) 1233. |
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233 | \end{enumerate} |
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234 | |
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235 | \end{htmlonly} |
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