Plot $X_0$ as a function of A and Z and put points for the materials in our simulation.
Plot $X_0$(in cm) as a function of A and Z and put points for the non-compound materials in our simulation.
\item The number of radiation lengths an electromagnetic particle travels in a material until the shower maximum follows the relation
\begin{equation}
t_{max}\,\propto\,\mathrm{ln} E_0 / E_c \quad .
\end{equation}
Verify qualitatively the position of the shower maximum, $t_{max}$, for 3 - 4 choices of energies in simulation.
A rule of thumb for the length in which $99\%$ of the initial energy is deposited in the material is
\begin{equation}
L ( 99 \% ) = (t_{max} + 0.08 Z + 9.6)[X_0] \quad .
\end{equation}
\begin{itemize}
\item[a)] Verify qualitatively the position of $t_{max}$ for 3 - 4 choices of energies in simulation.
\item[b)]Verify qualitatively $L (99\%)$ in simulation for a 50 GeV shower. If you use PbW04 for you calorimeter, simply use the Z of tungsten here as approximation.
\item[a)] Verify qualitatively $L (99\%)$ in simulation for a 50 GeV shower. If you use PbW04 for you calorimeter, simply use the Z of tungsten here as approximation.
\item[b)]How many X$_0$ does this correspond to (a number to keep in mind)?
\item[c)] How many X$_0$ are need to capture $95\%$ of the initial energy? (Note: the simulation also includes processes such as ionisation / excitation / Compton Scattering / Rayleigh Scattering)
\end{itemize}
\item The Moliere radius describes the transversal expansion of an electromagnetic shower, mostly by low-energy electrons. What is the Moliere radius of Pb or PbWO$_4$? What does that mean for our calorimeter?
