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