Method
The G4eIonisation class provides the continuous and discrete
energy losses of electrons and positrons due to ionisation in a material
according to the approach described in Mean Energy Loss. The value of
the maximum energy transferable to a free electron \(T_{max}\) is
given by the following relation:
(125)\[\begin{split}T_{max} = \left\{ \begin{array}{ll}
E-mc^2 & {\mbox{for} \hspace{.2cm} e^+} \\
(E-mc^2)/2 & {\mbox{for} \hspace{.2cm} e^- } \\
\end{array} \right .\end{split}\]
where \(mc^2\) is the electron mass. Above a given threshold energy
the energy loss is simulated by the explicit production of delta rays by
Möller scattering (\(e^- e^-\)), or Bhabha scattering
(\(e^+ e^-\)). Below the threshold the soft electrons ejected are
simulated as continuous energy loss by the incident \({e^{\pm}}\).
Continuous Energy Loss
The integration of (31) leads to the Berger-Seltzer formula
[MC70]:
(126)\[\left. \frac{dE}{dx} \right]_{T < T_{cut}} =
2 \pi r_e^2 mc^2 n_{el} \frac{1}{\beta^2}
\left [\ln \frac{2(\gamma + 1)} {(I/mc^2)^2}+ F^{\pm} (\tau , \tau_{up})
- \delta \right ]\]
with
\[\begin{split}r_e &= \mbox{classical electron radius:} \quad e^2/(4 \pi \epsilon_0 mc^2 ) \\
mc^2 &= \mbox{mass energy of the electron} \\
n_{el} &= \mbox{electron density in the material} \\
I &= \mbox{mean excitation energy in the material}\\
\gamma &= E/mc^2 \\
\beta^2 &= 1-(1/\gamma^2) \\
\tau &= \gamma-1 \\
T_{cut} &= \mbox{minimum energy cut for } \delta \mbox{-ray production} \\
\tau_c &= T_{cut}/mc^2 \\
\tau_{max} &= \mbox{maximum energy transfer:} \tau \mbox{ for } e^+, \tau/2 \mbox{ for } e^- \\
\tau_{up} &= \min(\tau_c,\tau_{max}) \\
\delta &= \mbox{density effect function} .\end{split}\]
In an elemental material the electron density is
\[n_{el} = Z \: n_{at} = Z \: \frac{\mathcal{N}_{av} \rho}{A} .\]
\(\mathcal{N}_{av}\) is Avogadro’s number, \(\rho\) is the
material density, and \(A\) is the mass of a mole. In a compound
material
\[n_{el} = \sum_i Z_i \: n_{ati}
= \sum_i Z_i \: \frac{\mathcal{N}_{av} w_i \rho}{A_i} ,\]
where \(w_i\) is the proportion by mass of the \(i^{th}\)
element, with molar mass \(A_i\) .
The mean excitation energies \(I\) for all elements are taken from
[eal84].
The functions \(F^{\pm}\) are given by :
\[\begin{split}F^+ (\tau,\tau_{up}) &= \ln(\tau\tau_{up} )
-\frac{\tau_{up}^2}{\tau}\left[\tau + 2 \tau_{up} -
\frac{3\tau_{up}^2 y } {2} -\left(\tau_{up} - \frac{\tau_{up}^3 }{3} \right) y^2
- \left (\frac{\tau_{up}^2}{2} - \tau
\frac{\tau_{up}^3}{3} + \frac{\tau_{up}^4 } {4} \right)
y^3 \right] \\
F^- (\tau,\tau_{up} ) &= -1 -\beta^2
+\ln \left [(\tau - \tau_{up})
\tau_{up} \right ] + \frac{\tau}{\tau -\tau_{up}}
+ \frac{1}{\gamma^2} \left [
\frac{\tau_{up}^2}{2} + ( 2\tau +1) \ln
\left (1- \frac{\tau_{up}}{\tau} \right ) \right ]\end{split}\]
where \(y = 1/(\gamma+1)\).
The density effect correction is calculated according to the formalism
of Sternheimer [SP71b]:
\(x\) is a kinetic variable of the particle :
\(x = \log_{10}(\gamma \beta) = \ln(\gamma^{2} \beta^{2})/4.606\),
and \(\delta(x)\) is defined by
\[\begin{split}\begin{array}{rll}
\mbox{for} & x < x_0 : & \delta(x) = 0 \\
\mbox{for} & x \in [x_0,\ x_1] : & \delta(x) = 4.606 x - C + a(x_1 - x)^m \\
\mbox{for} & x > x_1 : & \delta(x) = 4.606 x - C
\end{array}\end{split}\]
where the matter-dependent constants are calculated as follows:
\[\begin{split}\begin{array}{lcl}
h\nu_p & = & \mbox{ plasma energy of the medium }
= \sqrt{4\pi n_{el} r_e^3} mc^2/\alpha
= \sqrt{4\pi n_{el} r_e} \hbar c \\
C & = & 1 + 2 \ln (I/h\nu_p) \\
x_a & = & C/4.606 \\
a & = & 4.606(x_a - x_0)/(x_1 - x_0)^m \\
m & = & 3 .
\end{array}\end{split}\]
For condensed media
\[\begin{split}\begin{array}{ll}
I < 100 \: \mbox{eV} & \left \{
\begin{array}{rll}
\mbox{for } C \leq 3.681 & x_0 = 0.2 & x_1 = 2 \\
\mbox{for } C > 3.681 & x_0 = 0.326 C - 1.0 & x_1 = 2
\end{array} \right . \\
I \geq 100 \: \mbox{eV} & \left \{
\begin{array}{rll}
\mbox{for } C \leq 5.215 & x_0 = 0.2 & x_1 = 3 \\
\mbox{for } C > 5.215 & x_0 = 0.326 C - 1.5 & x_1 = 3
\end{array} \right .
\end{array}\end{split}\]
and for gaseous media
\[\begin{split}\begin{array}{rlll}
\mbox{for} & C < 10. & x_0 = 1.6 & x_1 = 4 \\
\mbox{for} & C \in [10.0,\ 10.5[ & x_0 = 1.7 & x_1 = 4 \\
\mbox{for} & C \in [10.5,\ 11.0[ & x_0 = 1.8 & x_1 = 4 \\
\mbox{for} & C \in [11.0,\ 11.5[ & x_0 = 1.9 & x_1 = 4 \\
\mbox{for} & C \in [11.5,\ 12.25[ & x_0 = 2. & x_1 = 4 \\
\mbox{for} & C \in [12.25,\ 13.804[ & x_0 = 2. & x_1 = 5 \\
\mbox{for} & C \geq 13.804 & x_0 = 0.326 C -2.5 & x_1 = 5 .
\end{array}\end{split}\]
Total Cross Section per Atom and Mean Free Path
The total cross section per atom for Möller scattering (\(e^- e^-\))
and Bhabha scattering (\(e^+ e^-\)) is obtained by integrating
Eq. (32). In Geant4 \(T_{cut}\) is always 1 keV or larger. For
delta ray energies much larger than the excitation energy of the
material (\(T \gg I\)), the total cross section becomes
[MC70] for Möller scattering,
\[\sigma ( Z,E,T_{cut} ) = \frac {2 \pi r_e^2 Z}{\beta^2(\gamma -1)}
\left[\frac{(\gamma-1)^2} {\gamma^2}\left(\frac{1}{2}-x\right)
+\frac{1}{x}-\frac{1}{1-x}-\frac{2\gamma-1}{\gamma^2}\ln
\frac{1-x}{x}\right] ,\]
and for Bhabha scattering (\(e^+ e^-\)),
\[\sigma (Z,E,T_{cut}) = \frac{ 2 \pi r_e^2 Z }{(\gamma -1)}
\left [\frac {1 }{\beta^2} \left(\frac{1}{x}-1\right)
+ B_1 \ln x + B_2 (1-x) -
\frac {B_3 } {2} ( 1-x^2 ) +\frac{B_4}{3}(1-x^3)\right] .\]
Here
\[\begin{split}\begin{array}{lcllcl}
\gamma & = & E/mc^2 &
B_1 & = & 2-y^2 \\
\beta^2 & = & 1-(1/\gamma^2) &
B_2 & = & (1-2y)(3+y^2 ) \\
x & = & T_{cut}/(E-mc^2) &
B_3 & = & (1-2y)^2+(1-2y)^3 \\
y & = & 1/(\gamma + 1) &
B_4 & = & (1-2y)^3 .
\end{array}\end{split}\]
The above formulas give the total cross section for scattering above
the threshold energies
\[T_{\rm Moller}^{\rm thr} =2T_{cut} \hspace{2cm} \mbox{and} \hspace{2cm}
T_{\rm Bhabha}^{\rm thr} = T_{cut} .\]
In a given material the mean free path is then
\[\begin{array}{lll}
\lambda = (n_{at} \cdot \sigma)^{-1} & \mbox{or} &
\lambda = \left( \sum_i n_{ati} \cdot \sigma_i \right)^{-1} .
\end{array}\]
Simulation of Delta-ray Production
Differential Cross Section
For \(T \gg I\) the differential cross section per atom becomes
[MC70] for Möller scattering,
(127)\[\frac{d\sigma }{d \epsilon } = \frac{2 \pi r_e^2 Z}{\beta^2 (\gamma -1)}
\left[ \frac{(\gamma -1 )^2} {\gamma^2 }+\frac{1}{\epsilon}
\left(\frac{1}{\epsilon}-\frac{2 \gamma -1 } {\gamma^2 } \right) +
\frac{1}{1- \epsilon}\left(\frac{1} {1- \epsilon} - \frac{2 \gamma - 1}
{\gamma^2 }\right) \right]\]
and for Bhabha scattering,
(128)\[\frac{d \sigma}{d \epsilon}=\frac{2 \pi r_e^2 Z}{(\gamma -1)}\left[
\frac{1} {\beta^2 \epsilon^2}-\frac{B_1}{\epsilon}+B_2 - B_3 \epsilon
+ B_4 \epsilon^2\right] .\]
Here \(\epsilon = T/(E-mc^2)\). The kinematical limits of
\(\epsilon\) are
\[\epsilon_0 = \frac{T_{cut}}{E-mc^2} \leq \epsilon \leq \frac{1}{2}
\hspace{.2cm} \mbox{ for $e^- e^-$} \hspace{2cm}
\epsilon_0 = \frac{T_{cut}}{E-mc^2} \leq \epsilon \leq 1
\hspace{.2cm} \mbox{ for $e^+ e^-$} .\]
Sampling
The delta ray energy is sampled according to methods discussed in
Monte Carlo Methods. Apart from normalization, the cross section can be
factorized as
\[\frac{d\sigma}{d\epsilon}=f(\epsilon) g(\epsilon) .\]
For \(e^- e^-\) scattering
\[\begin{split}f(\epsilon) &= \frac{1}{\epsilon^2} \frac{\epsilon_0 }{1- 2\epsilon_0} \\
g(\epsilon) &= \frac{4}{9\gamma^2 - 10 \gamma + 5}\left[(\gamma -1)^2
\epsilon^2 - (2 \gamma^2 +2\gamma -1) \frac{\epsilon} {1- \epsilon }+
\frac{\gamma^2}{(1- \epsilon )^2 }\right]\end{split}\]
and for \(e^+ e^-\) scattering
\[\begin{split}f(\epsilon) &= \frac{1}{\epsilon^2} \frac{\epsilon_0}{1- \epsilon_0 } \\
g(\epsilon) &= \frac{B_0 -B_1 \epsilon +B_2 \epsilon^2
-B_3 \epsilon^3 +B_4 \epsilon ^4}{B_ 0-B_1\epsilon_0
+B_2\epsilon_0^2
-B_3 \epsilon_0^3 +B_4 \epsilon_0^4} .\end{split}\]
Here \(B_0=\gamma^2/(\gamma^2-1)\) and all other quantities have
been defined above.
To choose \(\epsilon\), and hence the delta ray energy,
\(\epsilon\) is sampled from \(f(\epsilon)\)
the rejection function \(g(\epsilon)\) is calculated using the
sampled value of \(\epsilon\)
\(\epsilon\) is accepted with probability \(g(\epsilon)\).
After the successful sampling of \(\epsilon\), the direction of the
ejected electron is generated with respect to the direction of the
incident particle. The azimuthal angle \(\phi\) is generated
isotropically and the polar angle \(\theta\) is calculated from
energy-momentum conservation. This information is used to calculate the
energy and momentum of both the scattered incident particle and the
ejected electron, and to transform them to the global coordinate system.
Bibliography