source: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/electromagnetic/utils/densityeffect.tex @ 1288

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$x$ is a kinetic variable of the particle :
 $ x = \log_{10}(\gamma \beta) = \ln(\gamma^{2} \beta^{2})/4.606 $, \linebreak 
 and $\delta(x)$ is defined by
\begin{equation}
% \label{muion.de1}
\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{equation}
where the matter-dependent constants are calculated as follows:
\begin{equation}
% \label{muion.de2}
\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{equation}
For condensed media
$$
\begin{array}{ll}
I < 100 \: \mbox{eV} & \left \{
\begin{array}{rll}
\mbox{for } C \leq 3.681 & x_0 = 0.2           & x_1 = 2 \\
\mbox{for } C >    3.681 & x_0 = 0.326 C - 1.0 & x_1 = 2
\end{array} \right . \\
I \geq 100 \: \mbox{eV} & \left \{
\begin{array}{rll}
\mbox{for } C \leq 5.215 & x_0 = 0.2           & x_1 = 3 \\
\mbox{for } C >    5.215 & x_0 = 0.326 C - 1.5 & x_1 = 3
\end{array} \right .
\end{array}
$$
and for gaseous media 
\[
\begin{array}{rlll}
\mbox{for} & C < 10.                  & x_0 = 1.6 & x_1 = 4  \\
\mbox{for} & C \in [10.0,\  10.5[     & x_0 = 1.7 & x_1 = 4  \\
\mbox{for} & C \in [10.5,\  11.0[     & x_0 = 1.8 & x_1 = 4  \\
\mbox{for} & C \in [11.0,\  11.5[     & x_0 = 1.9 & x_1 = 4  \\
\mbox{for} & C \in [11.5,\  12.25[    & x_0 = 2.  & x_1 = 4  \\
\mbox{for} & C \in [12.25,\  13.804[  & x_0 = 2.  & x_1 = 5  \\
\mbox{for} & C \geq 13.804            & x_0 = 0.326 C -2.5 & x_1 = 5 .
\end{array}
\]