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\bfseries Marie Curie and HGF-Alliance Training Event 4-6 Mar 2010\\
{\red Geant4 Simulation and ROOT Analysis of a Si Beam Telescope} 
\end{minipage}
}
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\vspace{2mm}
%
%\normalsize

%\noindent
{\bfseries \blue \LARGE \underline{2. Analysis Tasks}}
\vspace{4mm}

\noindent
\begin{minipage}{16cm}
\renewcommand{\baselinestretch}{0.3}
 Note: When we talk about the\\[-11mm] 
 \begin{itemize}\itemsep-2.5mm
 \item[-]
 spread or Root Mean Square (RMS) value of a distribution we mean the
 standard deviation of the distribution.
 \item[-]
 resolution (or $\sigma$) of an observable we also mean
 the standard deviation of the distribution: \mbox{reconstructed - true}.
 \end{itemize}
\end{minipage}
\vspace{2mm}  


\noindent 
{\bfseries Task 2.1: Position reconstruction} 
\begin{enumerate}[a)]
\item determine pedestal and noise (from noise file)
\item plot signal distributions of first five events
% clear what is meant by events?
% sometimes we say plot, sometimes histogram blabla.. harmonize?
\item implement cluster search algorithm with three signal/noise (S/N)
  threshold cuts, using S/N(strip)=2, S/N(seed strip)=3 and S/N(cluster)=5,\\
  plot number of cluster per event and number of strips per cluster
\item[d)]
implement hit position reconstruction using centre-of-gravity of cluster algorithm (plot position in units of strip number)
% histogram? control plots
\end{enumerate}

\noindent 
{\bfseries Task 2.2: Cluster charge distributions 
{\small (Energy loss vs $p$ 
and Landau distributions)}} 

\noindent
for $p_{beam} = (2,5,10,200)$ GeV/c:
\begin{enumerate}[a)]
\item
histogram cluster charge distribution for the different momenta
\item 
perform Landau type fit to the cluster charge distributions
\item
determine for the cluster charge distributions:
most probable value, mean and median energy loss and energy spread 
for the different momenta
% Note: here we say explicitly for the different momenta but for the other tasks
% usually not?
\end{enumerate}

%\newpage
\noindent 
{\bfseries Task 2.3: Momentum dependence of position and
angular resolution}

\noindent
for $p_{beam} = (2,5,10,200)$ GeV/c:
\begin{enumerate}[a)]
\item 
determine single hit resolutions $\sigma_{hit}$: histogram residua
reconstructed hit - true track positions at sensor planes
and note down the RMS values (may plot vs momenta)
\item 
compare measured vs expected hit position, for the latter use the true track
parameters at z=0 and extrapolate the track to the sensor planes (thus neglecting
multiple scattering effects). Histogram the corresponding residua and check
if they agree with expectations: 
\begin{itemize}
\item[-]
mean of zero (within the estimated error) and
\item[-]
$\mbox{spread}^2 = A +B/p^2$, determine A and B and compare 
to back on the envelope calculations
of the multiple scattering effects
\end{itemize}
\item
This task only for the 200 GeV/c $p_{beam}$ data in order to suppress multiple
scattering effects: determine single hit resolution $\sigma_{hit}$ 
from the reconstructed data only, 
using the triplet method: $ res = 1/2 \cdot (x_1+x_3) - x_2$, 
where the $x_i$ refer to the measured
cluster positions in the three  sensors. 
The spread of the residuum distribution should
be equal to $\sqrt{3/2}\,\sigma_{hit}$ 
\item
perform a straight line fit $ x = a_0 + a_1\cdot z$ to the three reconstructed hits
($\sigma$ from 2.3c) for $p_{beam}=200$~GeV/c. 
Histogram the fit parameter resolution distributions: 
$a_{0,fit} - a_{0,true}$ and  $a_{1,fit} - a_{1,true}$.
Note down mean and spread values. 
Compare the observed spreads with the fiterrors 
(output by the fit) 
of $a_0$ and $a_1$. \emph{Optional task:} Repeat this analysis for other $p_{beam}$ values. 
How does it change and why?
\end{enumerate}

\noindent
{\bfseries Task 2.4: Position resolution of the DUT vs rotation angle $\theta_{DUT}$}

\noindent
for $p_{beam}=200$ GeV/c only:\\
The DUT sensor replaces the middle plane of the telescope. 
%It should
%be assumed that the accuracy of the track prediction by the outer two telescope
%planes does not change (not correct for finite rotation angles because of the change
%of the effective thickness)
%
\begin{enumerate}[a)]
\item
histogram the cluster charge for the different DUT rotation angles $\theta_{DUT}$,
determine the mean values and compare the results to expectations
\item determine the hit resolution vs $\theta_{DUT}$ from the
difference true - measured
\item determine the hit resolution vs $\theta_{DUT}$ using the 
triplet method:\\ $ res = 1/2 \cdot(x_1 +x_3) - x_{DUT}$ with $\sigma_{1/3}$ from 2.3c
\item
determine how the hit position resolution depends on the cluster charge $Q$ (e.g. by comparing the
data $Q>Q_{median}$ with $Q<Q_{median}$)
\end{enumerate}

\newpage
\subsection*{Reminder: Definition of the detector and beam}
%
\normalsize
Coordinate system: z along beam,  (x,y) = (horizontal, vertical)
%
%\hrule 
\vspace{2mm}

{\bfseries Si beam telescope: 3 x-planes (i.e. stripes in y-direction):}\\[-7mm]
\begin{itemize}\itemsep-1.5mm
\item sensor thickness: 300 $\mu$m
\item no. strips: 48
\item strip pitch: 20 $ \mu$m
\item strip length: 10 mm
\item conversion energy-loss to no.electrons: 3.6 eV/e
\item rms noise (Gauss): 1000 e
\item cross talk: none
\item z-positions: $z_{sensor}=(200,600,1000)$ mm
\end{itemize}
\vspace{2mm}

{\bfseries Device under test (DUT): 1 x-plane can be rotated around y axis:}\\[-7mm]
\begin{itemize}\itemsep-1.5mm
\item sensor thickness: 300 $\mu$m
\item no. strips: 48
\item {\boldmath \bf strip pitch: 50 $\mu$m}
\item strip length: 10 mm
\item conversion energy-loss to no.electrons: 3.6 eV/e
\item rms noise (Gauss): 1000 e
\item {\boldmath \bf cross talk: 5\%}
\item z-positions: $z_{DUT}= 600$ mm (replacing the central plane of the beam telescope) 
\end{itemize}
\vspace{2mm}

{\bfseries Pion beam:}\\[-7mm]
\begin{itemize}\itemsep-1.5mm
\item mean position at $z=0$ mm: $\langle x_{beam} 
\rangle =  \langle y_{beam} \rangle = 0$ mm
\item spread position at $z=0$ mm: $\sqrt{\langle x_{beam}^2 \rangle} = 
\sqrt{\langle y_{beam}^2 \rangle} = 0.1$ mm
\item mean angle at $z=0$ mm: $\langle \theta_x \rangle = \langle \theta_y \rangle = 0$ mrad
\item spread angle at  $z=0$ mm: $\sqrt{\langle \theta_x^2 \rangle} = \sqrt{\langle \theta_y^2 \rangle} 
= 0.1$ mrad
\item $p_{beam} = (2,5,10,200)$ GeV/c
\end{itemize}
%\hrule
\vspace{2mm}

{\bfseries Sensor simulation:}\\[-7mm]
\begin{itemize}\itemsep-1.5mm
\item define strip volume (xyz): (strip pitch, strip length, sensor thickness)
\item calculate energy in MeV deposited in strip volume 
\item convert energy to charges (3.6 eV/e)
\item introduce cross talk (make sure that no charge is lost!)
\item introduce noise (Gauss with rms noise)
\end{itemize}

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