[1211] | 1 | |
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| 2 | \section[Bremsstrahlung]{Bremsstrahlung} \label{sec:em.ebrem} |
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| 3 | |
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| 4 | The class $G4eBremsstrahlung$ provides the energy loss of electrons and |
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| 5 | positrons due to the radiation of photons in the field of a nucleus |
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| 6 | according to the approach described in Section \ref{en_loss}. |
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| 7 | Above a given threshold energy the energy loss is simulated by the explicit |
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| 8 | production of photons. Below the threshold the emission of soft photons is |
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| 9 | treated as a continuous energy loss. |
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| 10 | |
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| 11 | %{\color{red} |
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| 12 | Below electron/positron energies of 1 GeV, the cross |
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| 13 | section evaluation is based on a dedicated parameterization, see |
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| 14 | Sec.\ \ref{sec:em.ebrem.param}. Above this limit an analytic cross |
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| 15 | section is used, cf.\ Sec.\ \ref{sec:em.ebrem.lpm}. |
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| 16 | %} |
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| 17 | In GEANT4 the Landau-Pomeranchuk-Migdal effect has also been implemented. |
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| 18 | |
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| 19 | \subsection{Cross Section and Energy Loss}\label{sec:em.ebrem.param} |
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| 20 | |
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| 21 | $d\sigma(Z,T,k)/dk$ is the differential cross section for the production of a |
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| 22 | photon of energy $k$ by an electron of kinetic energy $T$ in the field of an |
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| 23 | atom of charge $Z$. If $k_c$ is the energy cut-off below which the soft |
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| 24 | photons are treated as continuous energy loss, then the mean value of the |
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| 25 | energy lost by the electron is |
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| 26 | \begin{equation}\label{eq:ebrem.eloss} |
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| 27 | E_{Loss}^{brem} (Z,T,k_c ) = |
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| 28 | \int_{0}^{k_ c}k\frac{d \sigma (Z,T,k)}{dk}dk . |
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| 29 | \end{equation} |
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| 30 | The total cross section for the emission of a photon of energy larger than |
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| 31 | $k_c$ is |
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| 32 | \begin{equation}\label{eq:ebrem.discrete} |
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| 33 | \sigma_{brem} (Z,T,k_c ) = \int_{k_c}^{T}\frac{d \sigma (Z,T,k)}{dk} dk . |
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| 34 | \end{equation} |
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| 35 | \\ |
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| 36 | |
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| 37 | \subsubsection{Parameterization of the Energy Loss and Total Cross Section} |
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| 38 | |
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| 39 | The cross section and energy loss due to bremsstrahlung have been |
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| 40 | parameterized using the EEDL (Evaluated Electrons Data Library) data set |
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| 41 | \cite{eedl} as input. |
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| 42 | |
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| 43 | \noindent |
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| 44 | The following parameterization was chosen for the electron bremsstrahlung |
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| 45 | cross section : |
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| 46 | \begin{equation} |
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| 47 | \label{ebrem.a} |
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| 48 | \sigma (Z,T,k_c ) = Z(Z+\xi_{\sigma} ) (1-c_{sigh} Z^{1/4}) |
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| 49 | \left[ \ln \frac{T}{k_c} \right]^{\alpha} \frac{{f_s}}{N_{Avo}} |
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| 50 | \end{equation} |
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| 51 | where $f_s$ is a polynomial in $x = lg(T)$ with $Z$-dependent coefficients for |
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| 52 | $x < x_l$ , $f_s= 1 $ for $x \ge x_l$, $\xi_{\sigma}, c_{sigh}, \alpha$ are |
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| 53 | constants, $N_{Avo}$ is the Avogadro number. |
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| 54 | For the case of low energy electrons ($T \le T_{lim} = 10 MeV$) the above |
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| 55 | expression should be multiplied by |
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| 56 | |
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| 57 | \begin{equation} |
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| 58 | \left(\frac{T_{lim}}{T}\right)^{c_l} \cdot \left(1 + \frac {a_l}{\sqrt{Z} T}\right), |
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| 59 | \end{equation} |
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| 60 | with constant $c_l, a_l$ parameters. |
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| 61 | |
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| 62 | The energy loss parameterization is the following : |
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| 63 | |
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| 64 | \begin{equation} |
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| 65 | \label{ebrem.b} |
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| 66 | E_{Loss}^{brem} (Z,T,k_c ) =\frac{Z(Z+ \xi_l)(T+m)^2 } |
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| 67 | {(T+2m)}\left[\frac{k_c}{T}\right]^\beta (2-c_{lh} Z^{\frac{1}{4}} ) |
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| 68 | \frac{a + b \frac{T}{T_{lim}}}{1 + c \frac{T}{T_{lim}}} |
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| 69 | \frac{f_l}{N_{Avo}} |
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| 70 | \end{equation} |
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| 71 | where $m$ is the mass of the electron, $\xi_l, \beta, c_{lh}, a,b,c$ are |
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| 72 | constants, $f_l$ is a polynomial in $x = lg(T)$ with $Z$-dependent |
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| 73 | coefficients for $x < x_l$ , $f_l= 1 $ for $x \ge x_l$. |
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| 74 | For low energies this expression should be divided by |
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| 75 | |
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| 76 | \begin{equation} |
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| 77 | \left(\frac{T_{lim}}{T}\right)^{c_l} |
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| 78 | \end{equation} |
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| 79 | |
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| 80 | and if $T < k_c$ the expression should be multiplied by |
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| 81 | |
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| 82 | \begin{equation} |
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| 83 | \left(\frac{T}{k_c}\right)^{a_l} |
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| 84 | \end{equation} |
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| 85 | |
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| 86 | with some constants $c_l, a_l$. |
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| 87 | The numerical values of the parameters and the coefficients of the |
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| 88 | polynomyals $f_s$ and $f_l$ can be found in the class code. |
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| 89 | \\ |
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| 90 | |
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| 91 | \noindent |
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| 92 | The errors of the parameterizations (\ref{ebrem.a}) and (\ref{ebrem.b}) |
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| 93 | were estimated to be |
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| 94 | |
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| 95 | \begin{eqnarray*} |
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| 96 | \frac{\Delta\sigma} {\sigma} & = & \left \{ |
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| 97 | \begin{array}{llr} |
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| 98 | 6-8 \% & \mbox{for } & T \leq 1 MeV \\ |
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| 99 | \leq 4-5\% & \mbox{for } & 1 MeV < T |
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| 100 | \end{array} |
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| 101 | \right . \\[1cm] |
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| 102 | \frac{\Delta E_{Loss}^{brem}} |
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| 103 | {E_{Loss}^{brem}} & = & \left \{ |
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| 104 | \begin{array}{llr} |
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| 105 | 8 -10\% & \mbox{for } & T \leq1 MeV \\ |
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| 106 | 5-6\% & \mbox{for } & 1 MeV < T . |
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| 107 | \end{array} |
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| 108 | \right . |
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| 109 | \end{eqnarray*} |
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| 110 | |
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| 111 | |
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| 112 | \noindent |
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| 113 | When running GEANT4, the energy loss due to soft photon bremsstrahlung is |
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| 114 | tabulated at initialization time as a function of the medium and of the |
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| 115 | energy, as is the mean free path for discrete bremsstrahlung. |
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| 116 | |
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| 117 | \subsubsection{Corrections for $e^+ e^-$ Differences} |
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| 118 | |
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| 119 | The preceding section has dealt exclusively with electrons. One might expect |
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| 120 | that positrons could be treated the same way. According to reference |
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| 121 | \cite{ebrem.kim} however, the differences between the radiative loss of positrons |
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| 122 | and electrons are considerable and cannot be disregarded. |
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| 123 | The ratio of the radiative energy loss for positrons |
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| 124 | to that for electrons obeys a simple scaling law, is a |
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| 125 | function only of the quantity $T/Z^2$. \\ |
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| 126 | |
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| 127 | \noindent |
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| 128 | The radiative energy loss for electrons or positrons is given by |
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| 129 | \begin{eqnarray*} |
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| 130 | -\frac{1}{\rho} \left ( \frac{dE}{dx} \right )_{rad}^{\pm} & = & |
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| 131 | \frac{N_{Av} \alpha r_e^2}{A} (T+m) Z^2 \Phi_{rad}^{\pm}(Z,T) \\ |
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| 132 | \Phi^{\pm}_{rad}(Z,T) & = & \frac{1}{\alpha r_{e}^2 Z^2 (T+m)} |
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| 133 | \int^{T}_{0}{k\frac{d\sigma^{\pm}}{dk}dk} |
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| 134 | \end{eqnarray*} |
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| 135 | and it is the ratio |
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| 136 | \begin{eqnarray*} |
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| 137 | \eta & = & \frac{\Phi_{rad}^{+}(Z,T)}{\Phi_{rad}^{-}(Z,T)} = |
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| 138 | \eta \left (\frac{T}{Z^2}\right ) |
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| 139 | \end{eqnarray*} |
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| 140 | that obeys the scaling law. \\ |
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| 141 | |
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| 142 | \noindent |
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| 143 | The authors have calculated this function in the range $10^{-7} |
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| 144 | \leq \frac{T}{Z^2} \leq 0.5$, where the kinetic energy $T$ is expressed in |
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| 145 | MeV. Their {\it data} can be fairly accurately reproduced using a |
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| 146 | parametrization: |
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| 147 | |
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| 148 | \begin{eqnarray*} |
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| 149 | \eta & = & \left \{ |
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| 150 | \begin{array}{llr} |
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| 151 | 0 & \mbox{if } & x \leq -8 \\ |
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| 152 | \frac{1}{2} + \frac{1}{\pi} \arctan \left( a_1 x + a_3 x^3 |
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| 153 | + a_5 x^5 \right ) & \mbox{if } & -8 < x < 9 \\ |
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| 154 | 1 & \mbox{if } & x \geq 9 |
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| 155 | \end{array} |
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| 156 | \right . |
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| 157 | \end{eqnarray*} |
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| 158 | where |
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| 159 | \begin{eqnarray*} |
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| 160 | x & = & \log \left ( C \frac{T}{Z^2} \right ) \mbox{(T in GeV)} \\ |
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| 161 | C & = & 7.5221 \times 10^{6} \\ |
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| 162 | a_1 & = & 0.415 \\ |
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| 163 | a_3 & = & 0.0021 \\ |
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| 164 | a_5 & = & 0.00054 . |
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| 165 | \end{eqnarray*} |
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| 166 | |
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| 167 | |
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| 168 | \noindent |
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| 169 | The $e^+ e^-$ energy loss difference is not purely a low-energy phenomenon |
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| 170 | (at least for high $Z$), as shown in Table~\ref{ebrem.c}. |
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| 171 | |
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| 172 | \begin{table}[hbt] |
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| 173 | \begin{centering} |
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| 174 | \begin{tabular}{rr|r|r} \hline |
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| 175 | \multicolumn{1}{c}{$\frac{T}{Z^2} (GeV)$} |
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| 176 | & \multicolumn{1}{c|}{T} |
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| 177 | & \multicolumn{1}{c|}{$\eta$} |
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| 178 | & \multicolumn{1}{c}{$\left ( \frac{rad. \ loss}{total \ loss} |
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| 179 | \right )_{e^-}$} \\[3mm] \hline |
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| 180 | $10^{-9}$ & $\sim 7 keV$ & $\sim 0.1$ & $\sim 0\%$ \\ |
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| 181 | $10^{-8}$ & $67 keV $ & $\sim 0.2$ & $\sim 1\%$ \\ |
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| 182 | $2 \times 10^{-7}$ & $1.35 MeV$ & $\sim 0.5$ & $\sim 15\%$ \\ |
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| 183 | $2 \times 10^{-6}$ & $13.5 MeV$ & $\sim 0.8$ & $\sim 60\%$ \\ |
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| 184 | $2 \times 10^{-5}$ & $135. MeV$ & $\sim 0.95$ & $> 90\%$ \\ \hline |
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| 185 | \end{tabular} |
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| 186 | \caption{ratio of the $e^+ e^-$ radiative energy loss in lead |
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| 187 | (Z=82).} |
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| 188 | \label{ebrem.c} |
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| 189 | \end{centering} |
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| 190 | \end{table} |
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| 191 | |
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| 192 | |
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| 193 | \noindent |
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| 194 | The scaling property will be used to obtain the positron energy loss and |
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| 195 | discrete bremsstrahlung from the corresponding electron values. However, |
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| 196 | while scaling holds for the ratio of the total radiative energy losses, it |
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| 197 | is significantly broken for the photon spectrum in the screened case. That is, |
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| 198 | \begin{eqnarray*} |
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| 199 | \frac{\Phi^+}{\Phi^-} = \eta \left ( \frac{T}{Z^2} \right ) |
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| 200 | & \hspace{3cm} & |
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| 201 | \frac{\frac{d\sigma^+}{dk}}{\frac{d\sigma^-}{dk}} = |
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| 202 | \mbox{does not scale .} |
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| 203 | \end{eqnarray*} |
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| 204 | For the case of a point Coulomb charge, scaling would be restored for the |
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| 205 | photon spectrum. In order to correct for non-scaling, it is useful to note |
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| 206 | that in the photon spectrum from bremsstrahlung reported in \cite{ebrem.kim}: |
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| 207 | \begin{eqnarray*} |
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| 208 | \frac{d\sigma^{\pm}}{dk} = S^{\pm} \left( \frac{k}{T} \right ) |
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| 209 | \hspace{2cm} |
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| 210 | \frac{S^{+}(k)}{S^{-}(k)} \leq 1 & \hspace{1cm} & S^{+}(1) = 0 |
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| 211 | \hspace{2cm} S^{-}(1) > 0 |
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| 212 | \end{eqnarray*} |
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| 213 | One can further assume that |
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| 214 | \begin{eqnarray} |
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| 215 | \frac{d\sigma^+}{dk} = f(\epsilon) \frac{d\sigma^-}{dk} , |
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| 216 | & \hspace{2cm} & |
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| 217 | \epsilon = \frac{k}{T} |
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| 218 | \label{ebrem.d} |
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| 219 | \end{eqnarray} |
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| 220 | and require |
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| 221 | \begin{eqnarray} |
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| 222 | \int^{1}_{0}{f(\epsilon)d\epsilon} & = & \eta |
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| 223 | \label{ebrem.e} |
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| 224 | \end{eqnarray} |
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| 225 | in order to approximately satisfy the scaling law for the ratio of the total |
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| 226 | radiative energy loss. From the photon spectra the boundary conditions |
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| 227 | \begin{eqnarray} |
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| 228 | \left . |
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| 229 | \begin{array}{l} |
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| 230 | f(0) = 1 \\ |
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| 231 | f(1) = 0 |
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| 232 | \end{array} |
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| 233 | \right \} \hspace{2cm} \mbox{for all $Z,T$} |
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| 234 | \label{ebrem.f} |
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| 235 | \end{eqnarray} |
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| 236 | may be inferred. Choosing a simple function for $f$ |
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| 237 | \begin{eqnarray} |
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| 238 | f(\epsilon) & = & C (1-\epsilon)^{\alpha} \hspace{3cm} C,\alpha > 0 , |
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| 239 | \label{ebrem.g} |
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| 240 | \end{eqnarray} |
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| 241 | the conditions (\ref{ebrem.e}), (\ref{ebrem.f}) lead to: |
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| 242 | \begin{eqnarray*} |
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| 243 | C & = & 1 \\ |
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| 244 | \alpha & = & \frac{1}{\eta} - 1 \hspace{2cm} |
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| 245 | \mbox{($\alpha > 0$ because $\eta < 1$)} \\ |
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| 246 | f(\epsilon) & = & (1-\epsilon)^{\frac{1}{\eta}-1} . |
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| 247 | \end{eqnarray*} |
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| 248 | |
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| 249 | \noindent |
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| 250 | Now the weight factors $F_{l}$ and $F_{\sigma}$ for the positron continuous |
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| 251 | energy loss and the discrete bremsstrahlung cross section can be defined: |
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| 252 | |
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| 253 | \begin{eqnarray} |
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| 254 | F_{l} = \frac{1}{\epsilon_{0}} \int^{\epsilon_{0}}_{0} |
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| 255 | {f(\epsilon)d\epsilon} & \hspace{3cm} & |
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| 256 | F_{\sigma} = \frac{1}{1-\epsilon_{0}} \int^{1}_{\epsilon_{0}} |
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| 257 | {f(\epsilon)d\epsilon} |
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| 258 | \label{ebrem.h} |
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| 259 | \end{eqnarray} |
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| 260 | where $\epsilon_{0} = \frac{k_c}{T}$ and $k_c$ is the photon cut. In this |
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| 261 | scheme the positron energy loss and discrete bremsstrahlung can be calculated |
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| 262 | as: |
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| 263 | \begin{eqnarray*} |
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| 264 | \left ( - \frac{dE}{dx} \right )^{+} = F_{l} |
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| 265 | \left ( - \frac{dE}{dx} \right )^{-} & \hspace{2cm} & |
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| 266 | \sigma^{+}_{brems} = F_{\sigma} \sigma^{-}_{brems} |
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| 267 | \end{eqnarray*} |
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| 268 | |
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| 269 | \noindent |
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| 270 | In this approximation the photon spectra are identical, therefore the same |
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| 271 | sampling is used for generating $e^-$ or $e^+$ bremsstrahlung. The following |
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| 272 | relations hold: |
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| 273 | \begin{eqnarray*} |
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| 274 | F_{\sigma} & = & \eta (1-\epsilon_{0})^{\frac{1}{\eta}-1} |
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| 275 | < \eta \\ |
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| 276 | \epsilon_{0} F_{l} + (1-\epsilon_{0}) F_{\sigma} & = & \eta |
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| 277 | \hspace{6cm} \mbox{from the def (\ref{ebrem.h})} \\ |
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| 278 | \Rightarrow F_{l} & = & \eta \frac{1-(1-\epsilon_{0})^{\frac{1} |
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| 279 | {\eta}})}{\epsilon_{0}} > \eta \frac{1-(1-\epsilon_{0})} |
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| 280 | {\epsilon_{0}} = \eta \hspace{1cm} |
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| 281 | \Rightarrow \left \{ |
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| 282 | \begin{array}{l} |
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| 283 | F_{l} > \eta \\ |
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| 284 | F_{\sigma} < \eta |
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| 285 | \end{array} \right . |
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| 286 | \end{eqnarray*} |
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| 287 | which is consistent with the spectra. The effect of the difference in $e^-$ |
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| 288 | and $e^+$ bremsstrahlung can also be seen in electromagnetic shower |
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| 289 | development when the primary energy is not too high. |
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| 290 | |
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| 291 | |
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| 292 | %{\color{red} |
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| 293 | The dielectric suppression of the photon spectrum (density effect or |
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| 294 | TerMikaelian effect \cite{ebrem.migdal}) is governed by the suppression function |
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| 295 | \begin{equation} |
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| 296 | \label{eq:ebrem.density} |
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| 297 | S_p = \frac{k^2}{k^2 + {k_p}^2} , |
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| 298 | \end{equation} |
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| 299 | where the charakteristic energy $k_p$ is defined by |
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| 300 | \begin{equation} |
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| 301 | \label{eq:ebrem.o} |
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| 302 | k_p^2 = \frac{r_0 \lambda^2_e n}{\pi} \cdot E^2 . |
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| 303 | \end{equation} |
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| 304 | and the parameters are defined by |
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| 305 | \[ |
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| 306 | \begin{array}{ll} |
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| 307 | r_0 & \mbox{classical electron radius} \\ |
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| 308 | \lambda_e & \mbox{electron Compton wavelength} \\ |
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| 309 | n & \mbox{electron density in the material.} |
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| 310 | \end{array} |
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| 311 | \] |
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| 312 | %} |
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| 313 | |
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| 314 | \subsection{Simulation of Discrete Bremsstrahlung} |
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| 315 | |
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| 316 | The energy of the final state photons is sampled according to the spectrum |
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| 317 | \cite{ebrem.seltzer} of Seltzer and Berger. They have calculated the |
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| 318 | bremsstrahlung spectra for materials with atomic numbers Z = 6, 13, 29, 47, |
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| 319 | 74 and 92 in the electron kinetic energy range 1 keV - 10 GeV. Their tabulated |
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| 320 | results have been used as input in a fit of the parameterized function |
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| 321 | |
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| 322 | \[ |
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| 323 | S(x) = C k \frac{d \sigma}{d k} , |
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| 324 | \] |
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| 325 | which will be used to form the rejection function for the sampling process. |
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| 326 | The parameterization can be written as |
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| 327 | \begin{equation} |
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| 328 | \label{eq:phys341-1} |
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| 329 | S(x) = \left \{ |
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| 330 | \begin{array}{ll} |
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| 331 | (1-a_{h} \epsilon )F_{1}(\delta) + b_{h} \epsilon^{2} F_{2} (\delta) |
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| 332 | & T \geq 1 MeV \\ |
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| 333 | 1 + a_{l} x + b_{l} x^{2} & T < 1 MeV |
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| 334 | \end{array} \right . |
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| 335 | \end{equation} |
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| 336 | where |
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| 337 | \[ |
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| 338 | \begin{array}{lcl} |
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| 339 | C & & \mbox{normalization constant} \\ |
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| 340 | k & & \mbox{photon energy} \\ [1mm] |
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| 341 | T, E & & \mbox{kinetic and total energy of the primary electron} \\ |
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| 342 | x & = & \frac{k}{T} \\ [2mm] |
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| 343 | \epsilon & = & \frac{k}{E} = x \frac{T}{E} \\ |
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| 344 | \end{array} |
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| 345 | \] |
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| 346 | and $a_{h,l}$ and $b_{h,l}$ are the parameters to be fitted. The |
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| 347 | $F_{i}(\delta)$ screening functions depend on the screening variable |
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| 348 | \[ |
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| 349 | \begin{array}{lcll} |
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| 350 | \delta & = & \frac{136 m_{e}}{Z^{1/3} E} \frac{\epsilon}{1-\epsilon} \\ |
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| 351 | F_{1}(\delta) & = & F_{0} (42.392 - 7.796 \delta +1.961 \delta^{2} - F) |
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| 352 | & \delta \leq 1 \\ |
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| 353 | F_{2}(\delta) & = & F_{0} (41.734 - 6.484 \delta +1.250 \delta^{2} - F) |
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| 354 | & \delta \leq 1 \\ |
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| 355 | F_{1}(\delta) & = & F_{2}(\delta) = |
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| 356 | F_{0} (42.24 - 8.368 \ln(\delta + 0.952) -F) & \delta > 1 \\ |
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| 357 | F_{0} & = & \frac{1}{42.392-F} \\ |
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| 358 | F & = & 4 \ln Z - 0.55 (\ln Z)^{2} . |
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| 359 | \end{array} |
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| 360 | \] |
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| 361 | |
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| 362 | \noindent |
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| 363 | The ``high energy'' ($T >$ 1 MeV) formula is essentially the |
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| 364 | Coulomb-corrected, screened Bethe-Heitler formula (see e.g. |
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| 365 | \cite{ebrem.williams,ebrem.butcher,ebrem.egs4}). However, |
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| 366 | Eq.~(\ref{eq:phys341-1}) differs from Bethe-Heitler in two ways: |
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| 367 | \begin{enumerate} |
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| 368 | \item $a_{h}, b_{h}$ depend on $T$ and on the atomic number $Z$, whereas in |
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| 369 | the Bethe-Heitler spectrum they are fixed ($a_{h} = 1$, $b_{h} =0.75$); |
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| 370 | \item the function $F$ is not the same as that in the Bethe-Heitler |
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| 371 | cross-section; the present function gives a better behavior in the |
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| 372 | high frequency limit, i.e. when $k \rightarrow T$ ($x \rightarrow 1$). |
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| 373 | \end{enumerate} |
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| 374 | |
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| 375 | \noindent |
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| 376 | The $T$ and $Z$ dependence of the parameters are described by the equations: |
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| 377 | |
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| 378 | \begin{eqnarray*} |
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| 379 | a_{h} & = & 1 + \frac{a_{h1}}{u}+\frac{a_{h2}}{u^{2}}+\frac{a_{h3}}{u^{3}} \\ |
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| 380 | b_{h} & = & 0.75+\frac{b_{h1}}{u}+\frac{b_{h2}}{u^{2}}+\frac{b_{h3}}{u^{3}} \\ |
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| 381 | a_{l} & = & a_{l0} + a_{l1} u + a_{l2} u^{2} \\ |
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| 382 | b_{l} & = & b_{l0} + b_{l1} u + b_{l2} u^{2} \\ |
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| 383 | \mbox{with} \\ |
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| 384 | u & = & \ln \left ( \frac{T}{m_{e}} \right ) |
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| 385 | \end{eqnarray*} |
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| 386 | The parameters $a_{hi}, b_{hi}, a_{li}, b_{li}$ are polynomials of second order |
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| 387 | in the variable: |
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| 388 | |
---|
| 389 | \[ |
---|
| 390 | v = [Z (Z+1)]^{1/3} . |
---|
| 391 | \] |
---|
| 392 | In the limiting case $T \rightarrow |
---|
| 393 | \infty$, $a_{h} \rightarrow 1, b_{h} \rightarrow 0.75$, |
---|
| 394 | Eq.~(\ref{eq:phys341-1}) gives the Bethe-Heitler cross section. \\ |
---|
| 395 | |
---|
| 396 | \noindent |
---|
| 397 | There are altogether 36 linear parameters in the formulae and their values are |
---|
| 398 | given in the code. This parameterization reproduces the Seltzer-Berger tables |
---|
| 399 | to within 2-3 \% on average, with the maximum error being less than 10-12 \%. |
---|
| 400 | The original tables, on the other hand, agree well with the experimental data |
---|
| 401 | and theoretical (low- and high-energy) results ($<$ 10 \% below 50 MeV and |
---|
| 402 | $<$ 5 \% above 50 MeV). \\ |
---|
| 403 | |
---|
| 404 | \noindent |
---|
| 405 | Apart from the normalization the cross section differential in photon |
---|
| 406 | energy can be written as |
---|
| 407 | \[ |
---|
| 408 | \frac{d \sigma}{d k} = -\frac{g(x)}{x\ln x_{c}} = -\frac{S(x)}{S_{max}x\ln x_{c}}, |
---|
| 409 | \] |
---|
| 410 | where $x_{c} = k_{c}/T$ and $k_{c}$ is the photon cut-off energy below |
---|
| 411 | which the bremsstrahlung is treated as a continuous energy loss. Using this |
---|
| 412 | decomposition of the cross section and two random numbers $r_{1}$, $r_{2}$ |
---|
| 413 | uniformly distributed in $[0,1]$, the sampling of $x$ is done as follows: |
---|
| 414 | \begin{enumerate} |
---|
| 415 | \item |
---|
| 416 | sample $x$ from $-\frac{1}{x\ln x_{c}}$ setting $x = e^{r_{1} \ln x_{c}}$ |
---|
| 417 | \item calculate the rejection function $g(x)$ and: |
---|
| 418 | \begin{itemize} |
---|
| 419 | \item if $r_{2} > g(x)$ reject $x$ and go back to 1; |
---|
| 420 | \item if $r_{2} \leq g(x)$ accept $x$. |
---|
| 421 | \end{itemize} |
---|
| 422 | \end{enumerate} |
---|
| 423 | |
---|
| 424 | \noindent |
---|
| 425 | The application of the dielectric suppression \cite{ebrem.migdal} %and |
---|
| 426 | %the LPM effect |
---|
| 427 | requires that $\epsilon$ also be sampled. First, the rejection |
---|
| 428 | function must be multiplied by a suppression factor |
---|
| 429 | \[ |
---|
| 430 | C_M (\epsilon) =\frac{1 + C_0 / \epsilon_c^2} |
---|
| 431 | {1 + C_0 / \epsilon^2} |
---|
| 432 | \] |
---|
| 433 | where |
---|
| 434 | \[ |
---|
| 435 | C_0 =\frac{nr_0 \lambda^2 }{\pi}, \hspace{1cm} \epsilon_c = \frac{k_{c}}{E} |
---|
| 436 | \] |
---|
| 437 | \begin{itemize} |
---|
| 438 | \item[$n$] electron density in the medium |
---|
| 439 | \item[$r_0$] classical electron radius |
---|
| 440 | \item[$\lambda$] reduced Compton wavelength of the electron. |
---|
| 441 | \end{itemize} |
---|
| 442 | Apart from the Migdal correction factor, this is simply expression |
---|
| 443 | \ref{eq:ebrem.density} . This correction decreases the cross-section for low photon |
---|
| 444 | energies. \\ |
---|
| 445 | |
---|
| 446 | %\noindent |
---|
| 447 | %While sampling $\epsilon$, the suppression factor $f_{LPM}=\frac{S}{S_p}$ is |
---|
| 448 | %also used as a rejection function in order to take into account the LPM effect. |
---|
| 449 | %Here the supression factor is compared to a random number $r$ uniformly |
---|
| 450 | %distributed in the interval $[0,1]$. If $f_{LPM} \geq r$ the simulation |
---|
| 451 | %continues, otherwise the bremsstrahlung process concludes {\em without photon |
---|
| 452 | %production}. It can be seen that this procedure performs the LPM suppression |
---|
| 453 | %correctly. \\ |
---|
| 454 | |
---|
| 455 | \noindent |
---|
| 456 | After the successful sampling of $\epsilon$, the polar angles of the radiated |
---|
| 457 | photon are generated with respect to the parent electron's momentum. It is |
---|
| 458 | difficult to find simple formulae for this angle in the literature. For |
---|
| 459 | example the double differential cross section reported by |
---|
| 460 | Tsai~\cite{ebrem.tsai1,ebrem.tsai2} is |
---|
| 461 | \begin{eqnarray*} |
---|
| 462 | \frac{d \sigma}{dkd \Omega} |
---|
| 463 | & = & \frac{2 \alpha^{2}e^{2}}{\pi k m^{4}} |
---|
| 464 | \left\{ \left[ \frac{2\epsilon-2}{(1+u^2)^2}+ |
---|
| 465 | \frac{12u^2(1-\epsilon)}{(1+u^2)^4}\right] |
---|
| 466 | Z(Z+1) \right. \\ |
---|
| 467 | & & \mbox{} + \left. \left[ \frac{2-2\epsilon-\epsilon^{2}}{(1+u^2)^2}- |
---|
| 468 | \frac{4u^2(1-\epsilon)}{(1+u^2)^4} |
---|
| 469 | \right] |
---|
| 470 | \left[ X-2Z^{2}f_{c}((\alpha Z)^{2})\right] |
---|
| 471 | \right\} \\ |
---|
| 472 | u & = & \frac{E \theta}{m} \\ |
---|
| 473 | X & = & \int_{t_{min}}^{m^{2}(1+u^{2})^{2}} |
---|
| 474 | {\left [ G_{Z}^{el}(t) + G_{Z}^{in}(t) \right ] \frac{t-t_{min}} |
---|
| 475 | {t^{2}} dt} \\ |
---|
| 476 | G_{Z}^{el, in}(t) & & \mbox{atomic form factors} \\ |
---|
| 477 | t_{min} & = & \left [ \frac{k m^{2} (1+u^{2})}{2 E (E-k)} \right ] ^{2} |
---|
| 478 | = \left [ \frac{\epsilon m^{2} (1+u^{2})}{2 E (1-\epsilon)} \right ] ^{2} . |
---|
| 479 | \end{eqnarray*} |
---|
| 480 | The sampling of this distribution is complicated. It is also only an |
---|
| 481 | approximation to within a few percent, due at least to the presence of the |
---|
| 482 | atomic form factors. The angular dependence is contained in the variable |
---|
| 483 | $u = E \theta m^{-1}$. For a given value of $u$ the dependence of the shape |
---|
| 484 | of the function on $Z$, $E$ and $\epsilon = k/E$ is very weak. Thus, the |
---|
| 485 | distribution can be approximated by a function |
---|
| 486 | \begin{equation} |
---|
| 487 | f(u) = C \left( u e^{-au} + d u e^{-3au} \right) |
---|
| 488 | \end{equation} |
---|
| 489 | where |
---|
| 490 | \[ |
---|
| 491 | C = \frac{9a^{2}}{9 + d} \hspace{1cm} a = 0.625 \hspace{1cm} |
---|
| 492 | d = 27 |
---|
| 493 | \] |
---|
| 494 | where $E$ is in GeV. While this approximation is good at high energies, |
---|
| 495 | it becomes less accurate around a few MeV. However in that region the |
---|
| 496 | ionization losses dominate over the radiative losses. \\ |
---|
| 497 | |
---|
| 498 | \noindent |
---|
| 499 | The sampling of the function $f(u)$ can be done with three random numbers |
---|
| 500 | $r_i$, uniformly distributed on the interval [0,1]: |
---|
| 501 | \begin{enumerate} |
---|
| 502 | \item choose between $u e^{-au}$ and $d u e^{-3au}$: |
---|
| 503 | \[ |
---|
| 504 | b = \left \{ \begin{array}{ll} |
---|
| 505 | a & \mbox{if\hspace{0.5cm}}r_{1} < 9/(9+d) \\ |
---|
| 506 | 3a & \mbox{if\hspace{0.5cm}}r_{1} \geq 9/(9+d) |
---|
| 507 | \end{array} \right . |
---|
| 508 | \] |
---|
| 509 | \item sample $u e^{-bu}$: |
---|
| 510 | \[ |
---|
| 511 | u=-\frac{\log ( r_{2} r_{3}) }{b} |
---|
| 512 | \] |
---|
| 513 | \item check that: |
---|
| 514 | \[ |
---|
| 515 | u \leq u_{max} = \frac{E \pi}{m} |
---|
| 516 | \] |
---|
| 517 | otherwise go back to 1. |
---|
| 518 | \end{enumerate} |
---|
| 519 | The probability of failing the last test is reported in |
---|
| 520 | table~\ref{tb:phys341-1}. \\ |
---|
| 521 | |
---|
| 522 | \begin{table} |
---|
| 523 | \begin{centering} |
---|
| 524 | \begin{tabular}{|l|l|} |
---|
| 525 | \multicolumn{2}{c}{$\displaystyle |
---|
| 526 | P = \int^{\infty}_{u_{max}}{f(u) \: du} \hfill $} \\ [0.5cm] |
---|
| 527 | \hline |
---|
| 528 | E (MeV) & P(\%) \\ \hline |
---|
| 529 | 0.511 & 3.4 \\ |
---|
| 530 | 0.6 & 2.2 \\ |
---|
| 531 | 0.8 & 1.2 \\ |
---|
| 532 | 1.0 & 0.7 \\ |
---|
| 533 | 2.0 & $<$ 0.1 \\ \hline |
---|
| 534 | \end{tabular} |
---|
| 535 | \caption{Angular sampling efficiency} |
---|
| 536 | \label{tb:phys341-1} |
---|
| 537 | \end{centering} |
---|
| 538 | \end{table} |
---|
| 539 | |
---|
| 540 | |
---|
| 541 | \noindent |
---|
| 542 | The function $f(u)$ can also be used to describe the angular distribution of |
---|
| 543 | the photon in $\mu$ bremsstrahlung and to describe the angular distribution in |
---|
| 544 | photon pair production. \\ |
---|
| 545 | |
---|
| 546 | \noindent |
---|
| 547 | The azimuthal angle $\phi$ is generated isotropically. Along with $\theta$, |
---|
| 548 | this information is used to calculate the momentum vectors of the radiated |
---|
| 549 | photon and parent recoiled electron, and to transform them to the |
---|
| 550 | global coordinate system. |
---|
| 551 | The momentum transfer to the atomic nucleus is neglected. |
---|
| 552 | |
---|
| 553 | \subsection{Bremsstrahlung of high-energy electrons}\label{sec:em.ebrem.lpm} |
---|
| 554 | % |
---|
| 555 | %{\color{red} |
---|
| 556 | Above an electron energy of 1 GeV an analytic |
---|
| 557 | differential cross section representation is used |
---|
| 558 | \cite{ebrem.perl}, which was modified to account for the density |
---|
| 559 | effect and the Landau-Pomeranchuk-Migdal (LPM) effect |
---|
| 560 | \cite{ebrem.klein,ebrem.stanev}. |
---|
| 561 | %} |
---|
| 562 | |
---|
| 563 | \subsubsection{Relativistic Bremsstrahlung cross section} |
---|
| 564 | |
---|
| 565 | %{\color{red} |
---|
| 566 | The basis of the implementation is the well known high energy limit of |
---|
| 567 | the Bremsstrahlung process \cite{ebrem.perl}, |
---|
| 568 | \begin{multline} |
---|
| 569 | \frac{d\sigma}{dk} = \frac{4 \alpha r_e^2}{3k} \biggl[ \{ y^2 + 2 [ 1 |
---|
| 570 | + (1-y)^2 ] \} [ Z^2 ( F_{el} - f ) + Z F_{inel} ] \\+ (1-y) \frac{Z^2 |
---|
| 571 | + Z }{3} \biggr] |
---|
| 572 | \label{eq:perl} |
---|
| 573 | \end{multline} |
---|
| 574 | The {\em elastic from factor} $F_{el}$ and {\em inelastic form factor} |
---|
| 575 | $F_{inel}$, |
---|
| 576 | describe the scattering on the nucleus and on the shell electrons, |
---|
| 577 | respectively, and for $Z>4$ are given by \cite{ebrem.PDGreview} |
---|
| 578 | \begin{align*} |
---|
| 579 | F_{el} &= \log\biggl( \frac{184.15}{Z^{\frac{1}{3}} } |
---|
| 580 | \biggr) |
---|
| 581 | &\mbox{and}\quad |
---|
| 582 | F_{inel} &= \log\biggl( \frac{1194.}{Z^{\frac{2}{3}} } \biggr) \;. |
---|
| 583 | \end{align*} |
---|
| 584 | This corresponds to the complete screening approximation. |
---|
| 585 | The Coulomb correction is defined as \cite{ebrem.PDGreview} |
---|
| 586 | \begin{align*} |
---|
| 587 | f &= \alpha^2 Z^2 \sum_{n=1}^\infty |
---|
| 588 | \frac{1}{n ( n^2 + \alpha^2Z^2 )} |
---|
| 589 | \end{align*} |
---|
| 590 | % |
---|
| 591 | This approach provides an analytic differential cross section for an |
---|
| 592 | efficient evaluation in a Monte Carlo computer code. Note |
---|
| 593 | that in this approximation the differential cross section $d\sigma/dk$ |
---|
| 594 | is independent of the energy of the initial electron and is also valid |
---|
| 595 | for positrons. |
---|
| 596 | |
---|
| 597 | The total integrated cross section $\int d\sigma/dk \, dk$ is |
---|
| 598 | divergent, but the energy loss integral $\int k d\sigma/dk \, dk$ is finite. |
---|
| 599 | % |
---|
| 600 | This allows the usual separation into |
---|
| 601 | continuous enery loss, and discrete photon production according to |
---|
| 602 | Eqs.\ (\ref{eq:ebrem.eloss}) and (\ref{eq:ebrem.discrete}). |
---|
| 603 | |
---|
| 604 | |
---|
| 605 | |
---|
| 606 | %} |
---|
| 607 | |
---|
| 608 | \subsubsection{Landau Pomeranchuk Migdal (LPM) effect} |
---|
| 609 | |
---|
| 610 | %{\color{red} |
---|
| 611 | At higher energies matter effects become more and |
---|
| 612 | more important. In GEANT4 the two leading matter effects, the |
---|
| 613 | LPM effect and the dielectric suppresion (or Ter-Mikaelian effect), |
---|
| 614 | are considered. The analytic cross section representation, eq.\ |
---|
| 615 | \eqref{eq:perl}, provides the basis for the incorporation of these |
---|
| 616 | matter effects. |
---|
| 617 | %} |
---|
| 618 | |
---|
| 619 | The LPM effect (see for example \cite{ebrem.galitsky, ebrem.anthony, ebrem.hansen} ) is the |
---|
| 620 | suppression of photon production due to the multiple scattering of the |
---|
| 621 | electron. If an electron undergoes multiple scattering while traversing the |
---|
| 622 | so called ``formation zone'', the bremsstrahlung amplitudes from before and |
---|
| 623 | after the scattering can interfere, reducing the probability of bremsstrahlung |
---|
| 624 | photon emission (a similar suppression occurs for pair production). The |
---|
| 625 | suppression becomes significant for photon energies below a certain value, |
---|
| 626 | given by |
---|
| 627 | \begin{equation} |
---|
| 628 | \label{ebrem.k} |
---|
| 629 | \frac{k}{E} < \frac{E}{E_{LPM}} , |
---|
| 630 | \end{equation} |
---|
| 631 | where |
---|
| 632 | \[ |
---|
| 633 | \begin{array}{ll} |
---|
| 634 | k & \mbox{photon energy} \\ |
---|
| 635 | E & \mbox{electron energy} \\ |
---|
| 636 | E_{LPM} & \mbox{characteristic energy for LPM effect (depend on the medium).} |
---|
| 637 | \end{array} |
---|
| 638 | \] |
---|
| 639 | The value of the LPM characteristic energy can be written as |
---|
| 640 | \begin{equation} |
---|
| 641 | \label{eq:ebrem.elpm} |
---|
| 642 | E_{LPM} = \frac{\alpha m^2 X_0}{4 h c} , |
---|
| 643 | \end{equation} |
---|
| 644 | where |
---|
| 645 | \[ |
---|
| 646 | \begin{array}{ll} |
---|
| 647 | \alpha & \mbox{fine structure constant} \\ |
---|
| 648 | m & \mbox{electron mass} \\ |
---|
| 649 | X_0 & \mbox{radiation length in the material} \\ |
---|
| 650 | h & \mbox{Planck constant} \\ |
---|
| 651 | c & \mbox{velocity of light in vacuum.} |
---|
| 652 | \end{array} |
---|
| 653 | \] |
---|
| 654 | |
---|
| 655 | %{\color{red} |
---|
| 656 | \noindent |
---|
| 657 | At high energies (approximately above 1 GeV) the differential cross section including the |
---|
| 658 | Landau-Pomeranchuk-Migdal effect, can be expressed using an evaluation |
---|
| 659 | based on \cite{ebrem.migdal,ebrem.stanev,ebrem.klein} |
---|
| 660 | \begin{multline} |
---|
| 661 | \label{eq:ebrem.lpm} |
---|
| 662 | \frac{d\sigma}{dk} = \frac{4 \alpha r_e^2}{3k} \biggl[ \xi(s) |
---|
| 663 | \{ y^2 G(s) + 2 [ 1 + (1-y)^2 ] \phi(s) \} \\ |
---|
| 664 | \times [ Z^2 (F_{el} - f ) + Z F_{inel} ] + (1-y) \frac{Z^2 + Z }{3} \biggr] |
---|
| 665 | \end{multline} |
---|
| 666 | where LPM suppression functions are defined by \cite{ebrem.migdal} |
---|
| 667 | \begin{align} |
---|
| 668 | G(s) &= 24s^2 \biggl( \frac{\pi}{2} - \int_0^\infty |
---|
| 669 | e^{-st} \frac{\sin(st)}{\sinh(\frac{t}{2})} dt\biggr) |
---|
| 670 | \end{align} |
---|
| 671 | and |
---|
| 672 | \begin{align} |
---|
| 673 | \phi(s) &= 12s^2 \Biggl( -\frac{\pi}{2} + \int_0^\infty |
---|
| 674 | e^{-st} \sin(st) \sinh\Bigl(\frac{t}{2}\Bigr)\, dt\Biggr) |
---|
| 675 | \end{align} |
---|
| 676 | They can be piecewise approximated with simple analytic functions, see e.g.\ \cite{ebrem.stanev}. |
---|
| 677 | The suppression function $\xi(s)$ is recursively defined via |
---|
| 678 | \begin{align*} |
---|
| 679 | s = \sqrt{ \frac{k\,E_{\rm LPM} }{8 E(E-k) \xi(s)} } |
---|
| 680 | \end{align*} |
---|
| 681 | but can be well approximated using an algorithm introduced by \cite{ebrem.stanev}. |
---|
| 682 | % |
---|
| 683 | The material dependent characteristic energy $E_{\rm LPM}$ is defined |
---|
| 684 | in Eq.\ (\ref{eq:ebrem.elpm}) |
---|
| 685 | %\begin{align} |
---|
| 686 | % E_{\rm LPM} = X_0 \frac{\alpha\,m_e^2}{8\pi\hbar c} |
---|
| 687 | %\end{align} |
---|
| 688 | according to \cite{ebrem.anthony}. Note that this definition differs from |
---|
| 689 | other definition (e.g. \cite{ebrem.klein}) by a factor $\frac{1}{2}$. |
---|
| 690 | |
---|
| 691 | An additional multiplicative factor governs the dielectric suppression |
---|
| 692 | effect (Ter-Mikaelian effect) \cite{ebrem.terMikaelian1}. |
---|
| 693 | \begin{align*} |
---|
| 694 | S(k) = \frac{k^2}{k^2+k_p^2} |
---|
| 695 | \end{align*} |
---|
| 696 | The characteristic photon energy scale $k_p$ is given by the |
---|
| 697 | plasma frequency of the media, defined as |
---|
| 698 | \begin{align*} |
---|
| 699 | k_p = \hbar \omega_p \frac{E_e}{m_e c^2} = \frac{\hbar E_e}{m_e |
---|
| 700 | c^2}\cdot \sqrt{\frac{n_e e^2}{\epsilon_0 m_e} } |
---|
| 701 | \;. |
---|
| 702 | \end{align*} |
---|
| 703 | |
---|
| 704 | % |
---|
| 705 | %\begin{align*} |
---|
| 706 | % S_{LPM} = \sqrt{ \frac{k\,E_{LPM} }{E^2 } } |
---|
| 707 | %\end{align*} |
---|
| 708 | % |
---|
| 709 | |
---|
| 710 | Both suppression effects, dielectric suppresion and LPM effect, reduce |
---|
| 711 | the effective formation length of the photon, |
---|
| 712 | so the suppressions {\em do not simply multiply.} |
---|
| 713 | %For the total suppression |
---|
| 714 | %$S$ the following equation holds (see \cite{ebrem.galitsky}) |
---|
| 715 | |
---|
| 716 | A consistent treatment of the overlap region, where both suppression |
---|
| 717 | mechanism, was suggested by \cite{ebrem.terMikaelian2}. |
---|
| 718 | The algorithm garanties that the LPM suppression is turned off |
---|
| 719 | as the density effect becomes important. This is achieved by defining |
---|
| 720 | a modified suppression variable $\hat{s}$ via |
---|
| 721 | \begin{align*} |
---|
| 722 | \hat{s} &= s \cdot \biggl( 1 + \frac{k_p^2}{k^2} \biggr) |
---|
| 723 | \end{align*} |
---|
| 724 | and using $\hat{s}$ in the LPM suppression functions $G(s)$ and |
---|
| 725 | $\phi(s)$ instead of $s$ in Eq.\ (\ref{eq:ebrem.lpm}). |
---|
| 726 | |
---|
| 727 | %The presented algorithms could be translated into Monte Carlo code, |
---|
| 728 | %which proved to perform efficiently for moderately high energies (1 |
---|
| 729 | %GeV to 100 TeV). |
---|
| 730 | %} |
---|
| 731 | |
---|
| 732 | \subsection{Status of this document} |
---|
| 733 | 09.10.98 created by L. Urb\'an. \\ |
---|
| 734 | 21.03.02 modif in angular distribution (M.Maire) \\ |
---|
| 735 | 27.05.02 re-written by D.H. Wright \\ |
---|
| 736 | 01.12.03 minor update by V. Ivanchenko \\ |
---|
| 737 | 20.05.04 updated by L.Urban \\ |
---|
| 738 | 09.12.05 minor update by V. Ivanchenko \\ |
---|
| 739 | 15.03.07 modify definition of Elpm (mma) \\ |
---|
| 740 | 12.12.08 update LPM effect and relativistic Model \\ |
---|
| 741 | 03.12.09 correct total cross section, formula 3 (mma) |
---|
| 742 | |
---|
| 743 | \begin{latexonly} |
---|
| 744 | |
---|
| 745 | \begin{thebibliography}{99} |
---|
| 746 | \bibitem{eedl} |
---|
| 747 | S.T.Perkins, D.E.Cullen, S.M.Seltzer, UCRL-50400 Vol.31 |
---|
| 748 | \bibitem{ebrem.geant3} |
---|
| 749 | GEANT3 manual ,CERN Program Library Long Writeup W5013 (October 1994). |
---|
| 750 | \bibitem{ebrem.galitsky} |
---|
| 751 | V.M.~Galitsky and I.I.~Gurevich, Nuovo Cimento 32 (1964) 1820. |
---|
| 752 | \bibitem{ebrem.anthony} |
---|
| 753 | P.L. Anthony et al., |
---|
| 754 | Phys.\ Rev.\ D {\bf 56} (1997) 1373, |
---|
| 755 | SLAC-PUB-7413/LBNL-40054 (February 1997). |
---|
| 756 | \bibitem{ebrem.seltzer} |
---|
| 757 | S.M.Seltzer and M.J.Berger, % Nucl.Inst.Meth. 80 (1985) 12. |
---|
| 758 | Nucl.\ Inst.\ Meth.\ {\bf B12} (1985) 95. |
---|
| 759 | \bibitem{ebrem.egs4} W.R. Nelson et al.:The EGS4 Code System. |
---|
| 760 | {\em SLAC-Report-265 , December 1985 } |
---|
| 761 | \bibitem{ebrem.messel} |
---|
| 762 | H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970. |
---|
| 763 | \bibitem{ebrem.migdal} |
---|
| 764 | A.B. Migdal, Phys.\ Rev.\ 103 (1956) 1811. |
---|
| 765 | \bibitem{ebrem.kim} |
---|
| 766 | L.~Kim et al., Phys.\ Rev.\ A33 (1986) 3002. |
---|
| 767 | \bibitem{ebrem.williams} |
---|
| 768 | R.W.~Williams, Fundamental Formulas of Physics, vol.2., Dover Pubs. (1960). |
---|
| 769 | \bibitem{ebrem.butcher} |
---|
| 770 | J.C.~Butcher and H. Messel., Nucl.\ Phys.\ 20 (1960) 15. |
---|
| 771 | \bibitem{ebrem.tsai1} |
---|
| 772 | Y-S.~Tsai, Rev.\ Mod.\ Phys 46 (1974) 815. |
---|
| 773 | \bibitem{ebrem.tsai2} |
---|
| 774 | Y-S.~Tsai, Rev.\ Mod.\ Phys 49 (1977) 421. |
---|
| 775 | \bibitem{ebrem.PDGreview} |
---|
| 776 | C. Amsler et al., |
---|
| 777 | %{\em 2008 Review of Particle Physics}, |
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| 778 | Physics Letters B667, 1 (2008). |
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| 779 | \bibitem{ebrem.perl} |
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| 780 | M.L.~Perl, |
---|
| 781 | % {\em Notes on the Landau Pomeranchuk Migdal effect: experiment and |
---|
| 782 | % theory}, |
---|
| 783 | in Procede Les Rencontres de physique de la Valle D'Aoste, |
---|
| 784 | SLAC-PUB-6514. |
---|
| 785 | \bibitem{ebrem.klein} |
---|
| 786 | S.~Klein, |
---|
| 787 | % {\em Suppression of bremsstrahlung and pair production due to |
---|
| 788 | % environmental factors}, |
---|
| 789 | Rev.\ Mod.\ Phys.\ {\bf 71} (1999) 1501-1538. |
---|
| 790 | \bibitem{ebrem.stanev} |
---|
| 791 | T.~Stanev et.al., |
---|
| 792 | % {\em Development of ultrahigh-energy electromagnetic cascades in |
---|
| 793 | % water and lead including the Landau-Pomeranchuk-Migdal effect} |
---|
| 794 | Phys. Rev. D25 (1982) 1291. |
---|
| 795 | \bibitem{ebrem.hansen} |
---|
| 796 | H.D.~Hansen et al., |
---|
| 797 | % {\em Landau-Pomeranchuk-Migdal effect for multihundred GeV electrons}, |
---|
| 798 | Phys.\ Rev.\ D {\bf 69} (2004) 032001. |
---|
| 799 | \bibitem{ebrem.terMikaelian1} |
---|
| 800 | M.L.~Ter-Mikaelian, Dokl.\ Akad. Nauk SSSR 94 (1954) 1033. |
---|
| 801 | \bibitem{ebrem.terMikaelian2} |
---|
| 802 | M.L.~Ter-Mikaelian, |
---|
| 803 | {\em High-energy Electromagnetic Processes in Condensed Media}, |
---|
| 804 | Wiley, (1972). |
---|
| 805 | |
---|
| 806 | \end{thebibliography} |
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| 807 | |
---|
| 808 | \end{latexonly} |
---|
| 809 | |
---|
| 810 | \begin{htmlonly} |
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| 811 | |
---|
| 812 | \subsection{Bibliography} |
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| 813 | |
---|
| 814 | \begin{enumerate} |
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| 815 | \item S.T.Perkins, D.E.Cullen, S.M.Seltzer, UCRL-50400 Vol.31 |
---|
| 816 | \item GEANT3 manual ,CERN Program Library Long Writeup W5013 (October 1994). |
---|
| 817 | \item V.M.Galitsky and I.I.Gurevich. Nuovo Cimento 32 (1964) 1820. |
---|
| 818 | \item P.L. Anthony et al., Phys. Rev. D56 (1997) 1373, SLAC-PUB-7413/LBNL-40054 (February 1997) |
---|
| 819 | \item S.M.Seltzer and M.J.Berger. Nucl.Inst.Meth. 12 (1985) 95. |
---|
| 820 | \item W.R. Nelson et al.:The EGS4 Code System. |
---|
| 821 | {\em SLAC-Report-265 , December 1985 } |
---|
| 822 | \item H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970. |
---|
| 823 | \item A.B. Migdal. Phys.Rev. 103. (1956) 1811. |
---|
| 824 | \item L. Kim et al. Phys. Rev. A33 (1986) 3002. |
---|
| 825 | \item R.W. Williams, Fundamental Formulas of Physics, vol.2., Dover Pubs. (1960). |
---|
| 826 | \item J. C. Butcher and H. Messel. Nucl.Phys. 20. (1960) 15. |
---|
| 827 | \item Y-S. Tsai, Rev. Mod. Phys. 46. (1974) 815. |
---|
| 828 | \item Y-S. Tsai, Rev. Mod. Phys. 49. (1977) 421. |
---|
| 829 | \item |
---|
| 830 | C. Amsler et al., |
---|
| 831 | %{\em 2008 Review of Particle Physics}, |
---|
| 832 | Physics Letters B667, 1 (2008). |
---|
| 833 | \item |
---|
| 834 | M.L.~Perl, |
---|
| 835 | in Procede Les Rencontres de physique de la Valle D'Aoste, |
---|
| 836 | SLAC-PUB-6514. |
---|
| 837 | \item |
---|
| 838 | S.~Klein, |
---|
| 839 | Rev.\ Mod.\ Phys.\ {\bf 71} (1999) 1501-1538. |
---|
| 840 | \item |
---|
| 841 | T.~Stanev et.al., |
---|
| 842 | Phys. Rev. D25 (1982) 1291. |
---|
| 843 | \item |
---|
| 844 | H.D.~Hansen et al., |
---|
| 845 | Phys.\ Rev.\ D {\bf 69} (2004) 032001. |
---|
| 846 | \item |
---|
| 847 | M.L.~Ter-Mikaelian, Dokl.\ Akad. Nauk SSSR 94 (1954) 1033. |
---|
| 848 | \item |
---|
| 849 | M.L.~Ter-Mikaelian, |
---|
| 850 | {\em High-energy Electromagnetic Processes in Condensed Media}, |
---|
| 851 | Wiley, (1972). |
---|
| 852 | |
---|
| 853 | \end{enumerate} |
---|
| 854 | |
---|
| 855 | \end{htmlonly} |
---|