\section{Electromagnetic shower measurements}
\label{sec:showers}

EM showers were created in the EUDET telescope by inserting tungsten sheets
between the two telescope arms. The positron data taken for each beam momentum
and amount of tungsten were shown in
table~\ref{tab::telescope:dataset}.
Clearly, with the tungsten only inserted in one position, and using sensors
which are limited in size to around $2 \times 1$\,cm$^2$, it is not possible
to contain the shower and make an energy resolution measurement directly.
The aim of this study is to measure the properties of the EM shower at
very high granularity. In particular, the main aim was to measure
the shower density as a function of
the angle of the particles to the shower axis.

ERROR ON ANGLE DUE TO THICKNESS OF TUNGSTEN?

\subsection{Measurement of shower density}

The density of particles in an EM shower falls off rapidly with distance
from the core of the shower. Therefore, to measure the density, it is
essential to estimate the position of the shower axis event by event.
To do this, the first arm, containing the first three layers of sensors, of the
telescope were used. Tracks were constructed by requiring a cluster in each
of the three layers with a chi-squared probability for the track fit of
greater than 0.05. In this case, since beam momenta down to 10\,GeV were
used, multiple scattering effects are no longer negligible. To allow for this,
the cluster position error in layer~0 was
inflated by adding an extra term in quadrature of $XXX/p_b$, where $p_b$ is
the beam momentum.

EPI UPSTREAM OR DOWNSTREAM?

The selected events were also vetoed for early showering
positrons by requiring the number of clusters in the first three layers to
be within three standard deviations of the average number (CHECK). In addition,
all combinations of the the remaining clusters in the first three layers 
after the track was
reconstucted were used to try to find a second track. Any event with a
second track satisfying a chi-squared probability for the track fit of
greater than 0.01 were discarded (CHECK).

CONVERT POSITION TO ANGLE. INDEPENDENT MEASUREMENT FOR EACH LAYER.
For the remaining events, the reconstructed track was projected onto each of
the last three sensors in turn. The positions of all clusters in these layers
was compared with the extrapolated track positions. The distribution of the
distances between the cluster and track projection gives an estimate of the
shower density as a function of the radial distance from the core of the
shower.

Since the sensors are relatively small compared with the beam width, then
it is possible to have showers close to the edge of the sensors,
where many particles miss the sensors and so
are unmeasured. A event by event
correction must be made in order that the density is not
biased by such cases. For each event, the active sensor area available for
measuring particles around the track projection was recorded as a function
of the distance from the track projection. This gives a normalisation
function with which to divide the observed cluster distribution so as to
get a true cluster density as a function of distance from the track.

Figure~\ref{fig:shower:example} shows the distribution of this angle for
the example case of the beam momentum of 100\,GeV and $99.9X_0$ of tungsten
as measured in layer~4.
This distribution can be characterised by the peak height, which gives the
core density at zero angle, and the width.
The dependences of the peak height and width for all combinations of
beam momentum and tungsten thickness are shown in
figure~\ref{fig:shower:angle}.
It is seen that consistent results are obtained from all three sensor
layers.

\subsection{Comparison with simulation}

Can only effectively measure charged particle contribution. Much lower
rate of photon interactions with sensor.

GEANT4 EM SHOWER GENERATORS?

%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=8cm]{showers/pulse_exp_shaper3b.png}
\caption{\sl 
Simulated reponse of pre-shape circuit to pulses of the same magnitude
but with varying collection times. The pulse charge is injected at 13\,$\mu$s on
the time axis and
collected with an exponential rise, with time constants of 0\,ns (black, dashed),
25\,ns (green), 50\,ns (blue) and 75\,ns (red). Zero corresponds to
a diode interaction while a typical epitaxial interaction corresponds
to the 50 or 75\,ns curve.
}
\label{fig:simulation:time}
\end{center}
\end{figure}

\subsection{Track-independent method}

$\Delta r_{ij}$ for all pairs of clusters $i$ and $j$. Independent of track
reconstruction but sensitive to core density for small values of $\Delta r$.
EXAMPLE PLOT; CHARACTERISE BY PEAK AND WIDTH. COMPARE WITH SIM.
