\section{Bremsstrahlung}\label{lowebrems}

 The class G4LowEnergyBremsstrahlung calculates the continuous energy loss due to low energy gamma emission and
simulates the gamma production by electrons. 
 The gamma production threshold for a given material $\omega_c$ is used to separate the continuous and the 
discrete parts of the process. The energy loss of an electron with the incident energy $T$ are expressed
via the integrand over energy of the gammas:

\begin{equation}
{dE\over dx}=\sigma(T){{\int^{\omega_c}_{0.1eV}t{d\sigma\over d\omega}d\omega} \over{\int^{T}_{0.1eV}
{d\sigma\over d\omega}d\omega}},
\end{equation}

 where $\sigma(T)$ is the total cross-section at a given incident kinetic energy, $T$, $0.1eV$ is the low energy limit
of the EEDL data. The production cross-section is a complimentary function:

\begin{equation}
\sigma=\sigma(T){{\int^{T}_{\omega_c}{d\sigma\over d\omega}d\omega}\over {\int^{T}_{0.1eV}{d\sigma\over d\omega}d\omega}}.
\end{equation}

 The total cross-section, $\sigma_s$, is obtained from an interpolation of the evaluated cross-section data in the EEDL
library~\cite{io-EEDL}, according to the formula (\ref{eqloglog}) in Section~\ref{subsubsigmatot}.

 The EEDL data~\cite{br-leg4} of total cross-sections are parametrised~\cite{br-EEDL}  according to (\ref{eqloglog}).
The probability of the emission of a photon with energy, $\omega$, considering an electron of incident kinetic energy, 
$T$, is generated according to the formula:

\begin{equation}
\label{eqbrem}
{d\sigma \over d\omega} = {F(x) \over x}, \;\; \mbox{with} x = {\omega \over T}.
\end{equation}

 The function, $F(x)$, describing energy spectra of the outcoming photons is taken from the EEDL library. For each 
element 15 points in $x$ from 0.01 to 1 are used for the linear interpolation of this function. The function $F$ is 
normalised by the condition $F(0.01) = 1$. The energy distributions of the emitted photons available in the EEDL 
library are for only a few incident electron energies (about 10 energy points between 10 eV and 100 GeV). For other 
energies a logarithmic interpolation formula (\ref{eqloglog}) is used to obtain values for the function, $F(x)$.
For high energies, the spectral function is very close to:

\begin{equation}
  F(x) = 1 - x + 0.75x^2. 
\end{equation}

\subsection{Bremsstrahlung angular distributions}

 The angular distribution of the emitted photons with respect to the incident
electron can be sampled according to three alternative generators described below.
The direction of the outcoming electron is determined from the energy-momentum balance. 
This generators are currently implemented in G4ModifiedTsai, G4Generator2BS and 
G4Generator2BN classes. 

\subsubsection*{G4ModifiedTsai}

\noindent
 The angular distribution of the emitted photons is obtained from a 
simplified \cite{br-g3} formula based on the Tsai cross-section \cite{br-tsai},
which is expected to become isotropic in the low energy limit. 

\subsubsection*{G4Generator2BS}

 In G4Generator2BS generator, the angular distribution of the emitted photons is obtained 
from the 2BS Koch and Motz bremsstrahlung double differential cross-section \cite{br-KandM}:

\begin{eqnarray*}
d\sigma_k,_\theta & = & \frac{4Z^2 r_0^2}{137} \frac{dk}{k} ydy \left\{
\frac{16y^2E}{(y^2+1)^4E_0}-\right.{} \nonumber \\ 
& & \left.\frac{(E_0+E)^2}{(y^2+1)^2E_0^2} + 
\left[ \frac{E_0^2+E^2}{(y^2+1)^2E_0^2}- \frac{4y^2E}{(y^2+1)^4E_0}\right]ln M(y)\right\}
\end{eqnarray*}

\noindent
where $k$ the photon energy, $\theta$ the emission angle, $E_0$ and $E$ are the 
initial and final electron energy in units of $m_e c^2$, $r_0$ is the classical
electron radius and $Z$ the atomic number of the material. $y$ and $M(y)$ are
defined as:
\begin{eqnarray*}
y&=&E_0\theta \nonumber \\
\frac{1}{M(y)}&=&\left(\frac{k}{2E_0E}\right)^2+\left(\frac{Z^{1/3}}{111(y^2+1)}\right)^2
\end{eqnarray*}

 The adopted sampling algorithm is based on the sampling scheme developed by 
A. F. Bielajew et al. \cite{br-pirs}, and latter implemented in EGS4. In this sampling algorithm
only the angular part of 2BS is used, with the emitted photon energy, $k$, determined by 
GEANT4 $\left(\frac{d\sigma}{dk}\right)$ differential cross-section.


\subsubsection*{G4Generator2BN}

\noindent
The angular distribution of the emitted photons is obtained from the 2BN Koch and Motz bremsstrahlung 
double differential cross-section \cite{br-KandM} that can be written as:

\begin{eqnarray*}
d\sigma_k,_\theta & = & \frac{Z^2 r_0^2}{8\pi 137}\frac{dk}{k} \frac{p}{p_0} d\Omega_k
\left \{ \frac{8\sin^2\theta (2E_0^2-1)}{p_0^2\Delta_0^4}- \right.{} \nonumber \\ 
& & \left.\frac{2(5E_0^2+2EE_0+3)}{p_0^2\Delta_0^2}
- \frac{2(p_0^2-k^2)}{Q^2\Delta_0}+\frac{4E}{p_2^2\Delta_0}+\frac{L}{pp_0} \right.{} \nonumber \\ 
& & \left. \left[ \frac{4E_0\sin^2\theta(3k-p_0^2E)}{p_0^2\Delta^4} + \frac{4E_0^2(E_0^2+E^2)}
{p_0^2\Delta_0^2}+ \right.\right.{} \nonumber \\ 
& & \left.\left. \frac{2-2(E_0^2-3EE_0+E^2)}{p_0^2\Delta_0^2}+\frac{2k(E_0^2+EE_0-1)}
{p_0^2\Delta_0}\right] \right.{} \nonumber \\ 
& & \left. -\left(\frac{4\epsilon}{p\Delta0}\right) + \left(\frac{\epsilon^Q}{pQ}\right)
\left[\frac{4}{\Delta^2_0}-\frac{6k}{\Delta_0}-\frac{2k(p_0^2-k^2)}{Q^2\Delta_0}\right]\right \}
\end{eqnarray*}
\noindent in which:
\begin{eqnarray*}
L&=&\ln\left[\frac{EE_0-1+pp_0}{EE_0-1-pp_0}\right] \nonumber \\
\Delta_0&=&E_0-p_0\cos\theta                       \nonumber \\
Q^2&=&p_0^2+k^2-2p_0k\cos\theta                    \nonumber \\
\epsilon&=&\ln\left[\frac{E+p}{E-p}\right]  \qquad  \epsilon^Q=\ln\left[\frac{Q+p}{Q-p}\right]
\end{eqnarray*}

\noindent
where $k$ is the photon energy, $\theta$ the emission angle and $(E_0,p_0)$ and $(E,p)$ are the total 
(energy, momentum) of the electron before and after the radiative emission, all in units of $m_e c^2$.\\
 Since the 2BN cross--section is a 2-dimensional  non-factorized  distribution an 
acceptance-rejection technique was the adopted. For the 2BN distribution, two functions 
$g_1(k)$ and $g_2(\theta)$ were defined:

\begin{equation}
g_1(k) = k^{-b} \qquad\qquad g_2(\theta)=\frac{\theta}{1+c\theta^2}
\end{equation}

\noindent
such that:

\begin{equation}
Ag_1(k)g_2(\theta) \ge \frac{d\sigma}{dkd\theta}
\end{equation}

\noindent
where A is a global constant to be completed. Both functions have an analytical 
integral $G$ and an analytical inverse $G^{-1}$. The $b$ parameter of $g_1(k)$ was 
empirically tuned and set to $1.2$. For positive $\theta$ values, $g_2(\theta)$ has a maximum  
at $\frac{1}{\sqrt(c)}$. $c$ parameter controls the function global shape and it was 
used to tune $g_2(\theta)$ according to the electron kinetic energy.\\
To generate photon energy $k$ according to $g_1$ and $\theta$ according to $g_2$ the
inverse-transform method was used. The integration of these functions gives

\begin{equation}
G_1 = C_1 \int_{k_{min}}^{k_{max}} k'^{-b}dk' = C_1 \frac{k^{1-b}-k^{1-b}_{min}}{1-b}
\end{equation}

\begin{equation}
G_2 = C_2 \int_{0}^{\theta} \frac{\theta'}{1+c\theta'^2}d\theta'=C_2 \frac{\log(1+c\theta^2)}{2c}
\end{equation}

\noindent
where $C_1$ and $C_2$ are two global constants chosen to normalize the integral in the overall range 
to the unit. The  photon momentum $k$ will range from a minimum cut value $k_{min}$ (required to avoid 
infrared divergence) to a maximum value equal to the  electron  kinetic energy  $E_k$, while the polar 
angle ranges from 0 to $\pi$, resulting for $C_1$ and $C_2$:

\begin{equation}
C_1 = \frac{1-b}{E_k^{1-b}} \qquad\qquad C_2 = \frac{2c}{\log(1+c\pi^2)}
\end{equation}

\noindent
$k$ and $\theta$ are then sampled according to:

\begin{equation}
k = \left[ \frac{1-b}{C_1}\xi_1 + k_{min}^{1-b} \right] \qquad\qquad \theta = \sqrt{\frac{\exp\left(\frac{2c\xi_2}{C_1}\right)}{2c}}
\end{equation}

\noindent
where $\xi_1$ and $\xi_2$ are uniformly sampled in the interval (0,1). The event is accepted if:

\begin{equation}
uAg_1(k)g_2(\theta) \le \frac{d\sigma}{dkd\theta}
\end{equation}

\noindent
where $u$ is a random number with uniform distribution in (0,1). The $A$ and $c$ parameters were computed 
in a logarithmic grid, ranging  from 1 keV to 1.5 MeV with 100 points per decade. 
Since the $g_2(\theta)$ function has a maximum at $\theta = \frac{1}{\sqrt{c}}$, 
the $c$ parameter was  computed  using  the  relation $c=\frac{1}{\theta_{max}}$. At the point ($k_{min},\theta_{max}$) 
where $k_{min}$ is the $k$ cut value, the double differential cross-section has its maximum value, since it is
monotonically decreasing in $k$ and thus the global normalization parameter $A$ is estimated from the relation:

\begin{equation}
A g_1(k_{min})g_2({\theta_{max}})= \left(\frac{d^2\sigma}{dkd\theta}\right)_{max}
\end{equation}

\noindent
where $g_1(k_{min})g_2({\theta_{max}}) =  \frac{k_{min}^{-b}}{2\sqrt{c}}$.
Since $A$ and $c$ can only be retrieved for a fixed number of electron kinetic energies there exists the possibility that
$A g_1(k_{min})g_2({\theta_{max}})\le\left(\frac{d^2\sigma}{dkd\theta}\right)_{max}$ for a given $E_k$. This is a small 
violation that can be corrected introducing an additional multiplicative factor to the $A$ parameter, which was 
empirically determined to be 1.04 for the entire energy range.\\ 

\subsubsection*{Comparisons between Tsai, 2BS and 2BN generators}

The currently available generators can be used according to the user required 
precision and timing requirements. Regarding the energy range, validation results 
indicate that for lower energies ($\le$ 100 keV) there is a significant
deviation on the most probable emission angle between Tsai/2BS generators 
and the 2BN generator - Figure \ref{br-dist}. The 2BN generator maintains however a good agreement 
with Kissel data \cite{Kissel}, derived from the work of Tseng and co-workers \cite{Pratt},
and it should be used for energies between 1 keV and 100 keV \cite{IEEE}.
As the electron kinetic energy increases, the different distributions tend to overlap 
and all generators present a good agreement with Kissel data.

\begin{figure}[hbtp]
\begin{center}
\setlength{\unitlength}{0.0105in}%
\includegraphics[width=4.7cm]{electromagnetic/lowenergy/br-10kev.eps}%
\includegraphics[width=4.7cm]{electromagnetic/lowenergy/br-100kev.eps}%
\includegraphics[width=4.7cm]{electromagnetic/lowenergy/br-500kev.eps}%
\end{center}
\caption{Comparison of polar angle distribution of bremsstrahlung photons ($k/T=0.5$) for
10 keV ({\em left}) and 100 keV ({\em middle}) and 500 keV ({\em right}) electrons in silver, 
obtained with Tsai, 2BS and 2BN generator}
\label{br-dist}
\end{figure}

\noindent
In figure \ref{br-eff} the sampling efficiency for the different generators are presented. 
The sampling generation efficiency was defined as the ratio between the
number of generated events and the total number of trials. As energies increases the sampling efficiency 
of the 2BN algorithm decreases from 0.65 at 1 keV electron kinetic energy down to almost 0.35 at 1 MeV. 
For energies up to 10 keV the 2BN sampling efficiency is superior or equivalent to the one of the 
2BS generator. These results are an indication that precision simulation of low energy bremsstrahlung 
can be obtained with little performance degradation. For energies above 500 keV, Tsai generator can be
used, retaining a good physics accuracy and a sampling efficiency superior to the 2BS generator.
%
\begin{figure}[hbtp]
\begin{center}
\setlength{\unitlength}{0.0105in}%
\includegraphics[width=8cm]{electromagnetic/lowenergy/br-eff.eps}
\end{center}
\caption{Sampling efficiency for Tsai generator, 2BS and 2BN Koch and Motz generators.}
\label{br-eff}
\end{figure} 

\subsection{Status of the document}

\noindent
30.09.1999 created by Alessandra Forti\\
07.02.2000 modified by V\'eronique Lef\'ebure\\
08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia\\
05.12.2001 modified by Vladimir Ivanchenko\\
13.05.2002 modified by Vladimir Ivanchenko\\
24.11.2003 modified by Andreia Trindade, Pedro Rodrigues and Luis Peralta\\

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{br-leg4}
  ``Geant4 Low Energy Electromagnetic Models for Electrons and Photons",
   J.Apostolakis et al., CERN-OPEN-99-034(1999), INFN/AE-99/18(1999) 
\bibitem{br-EEDL} 
  %http://reddog1.llnl.gov/homepage.red/Electron.htm
  ``Tables and Graphs of Electron-Interaction Cross-Sections from 10~eV to 100~GeV Derived from
  the LLNL Evaluated Electron Data Library (EEDL), Z=1-100"
  S.T.Perkins, D.E.Cullen, S.M.Seltzer,
  UCRL-50400 Vol.31
\bibitem{br-g3}
  ``GEANT, Detector Description and Simulation Tool",
  CERN Application Software Group, CERN Program Library Long Writeup W5013 
\bibitem{br-tsai}    
  ``Pair production and bremsstrahlung of charged leptons",
  Y. Tsai, Rev. Mod. Phys., Vol.46, 815(1974), Vol.49, 421(1977)
\bibitem{br-KandM}
   ``Bremsstrahlung Cross-Section Formulas and Related Data",
  H. W. Koch and J. W. Motz, Rev. Mod. Phys., Vol.31, 920(1959)
\bibitem{br-pirs} 
   ``Improved bremsstrahlung photon angular sampling in the EGS4 code system'',
   A. F. Bielajew, R. Mohan and C.-S. Chui, Report NRCC/PIRS-0203 (1989)
\bibitem{Kissel}  
  ``Bremsstrahlung from electron collisions with neutral atoms'', 
   L. Kissel, C. A. Quarls and R. H. Pratt, At. Data  Nucl. Data Tables, Vol. 28, 382(1983)
\bibitem{Pratt} 
  ``Electron bremsstrahlung angular distributions in the 1-500 keV energy range'', 
   H. K. Tseng,  R. H. Pratt  and  C. M. Lee , Phys. Rev. A, Vol. 19, 187(1979)
\bibitem{IEEE}
   ``GEANT4 Applications and Developments for Medical Physics Experiments'',
   P. Rodrigues et al. IEEE 2003 NSS/MIC Conference Record
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item
  ``Geant4 Low Energy Electromagnetic Models for Electrons and Photons",
   J.Apostolakis et al., CERN-OPEN-99-034(1999), INFN/AE-99/18(1999) 
\item 
  %http://reddog1.llnl.gov/homepage.red/Electron.htm
  ``Tables and Graphs of Electron-Interaction Cross-Sections from 10~eV to 100~GeV Derived from
  the LLNL Evaluated Electron Data Library (EEDL), Z=1-100"
  S.T.Perkins, D.E.Cullen, S.M.Seltzer,
  UCRL-50400 Vol.31
\item
  ``GEANT, Detector Description and Simulation Tool",
  CERN Application Software Group, CERN Program Library Long Writeup W5013 
\item   
  ``Pair production and bremsstrahlung of charged leptons",
  Y. Tsai, Rev. Mod. Phys., Vol.46, 815(1974), Vol.49, 421(1977)
\item
   ``Bremsstrahlung Cross-Section Formulas and Related Data",
  H. W. Koch and J. W. Motz, Rev. Mod. Phys., Vol.31, 920(1959)
\item 
   ``Improved bremsstrahlung photon angular sampling in the EGS4 code system'',
   A. F. Bielajew, R. Mohan and C.-S. Chui, Report NRCC/PIRS-0203 (1989)
\item  
  ``Bremsstrahlung from electron collisions with neutral atoms'', 
   L. Kissel, C. A. Quarls and R. H. Pratt, At. Data  Nucl. Data Tables, Vol. 28, 382(1983)
\item 
  ``Electron bremsstrahlung angular distributions in the 1-500 keV energy range'', 
   H. K. Tseng,  R. H. Pratt  and  C. M. Lee , Phys. Rev. A, Vol. 19, 187(1979)
\item
   ``GEANT4 Applications and Developments for Medical Physics Experiments'',
   P. Rodrigues et al. IEEE 2003 NSS/MIC Conference Record
\end{enumerate}

\end{htmlonly}


