\chapter{Particle Physics and Particle Detectors}
\label{DETECTOR}

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\section{Introduction} 
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  The modern physicist working in elementary particle research is continuing a quest for 
  understanding of the basic structure of matter which has gone on for thousands of years.
  {\it And so on ....}

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\section{The ALEPH Detector}
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  ALEPH (Apparatus for LEP pHysics) was one of four detectors
  positioned around the LEP collider at CERN.
  It was in operation for six years in the study of 
  $e^{+}e^{-}$ collisions at Z resonance.
  The ALEPH project was the outcome of a collaboration of several
  hundred physicists and engineers from thirty universities and
  national laboratories, principally from Europe but also from China
  and the U.S.A. Detailed performance of the ALEPH detector 
  can be found in \cite{ALEPH_HANDBOOK} and \cite{ALEPH_DET}.

  ALEPH is a particle detector, covering as much of the $4\pi$ solid angle as
  possible.
  It is designed to measure the momenta of charged
  particles, to measure the energy deposited in calorimeters by charged and
  neutral particles, to identify the three lepton flavors, 
  and to measure the distance of travel of short-lived particles such as the
  tau lepton and the b and c hadrons. Particular emphasis has been given to
  momentum resolution up to the highest energies (by means of a large tracking
  system in a 1.7 \unt{T} magnetic field), to electron identification (by means of a
  high segmented, projective electromagnetic calorimeter, as well as ionization
  measurement in the tracing system), and muon identification (with continuous
  tracing inside sufficient iron absorber to eliminate the hadrons).

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\subsection{\label{ALEPH_OVER} Overview}
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  ALEPH uses a coordinate system usually expressed in terms of
  Cartesian $(x,y,z)$ or cylindrical $(r,\phi,z)$ 
  coordinates. In both cases, the $z$ direction 
  is along the beam line with the $e^-$ direction defining the positive 
  axis. The positive $x$ direction points to the centre of LEP and 
  the positive $y$ direction is defined such that ($x,y,z$) forms a
  right-handed coordinate system.

  The tracking system involves three subdetectors:
  a silicon vertex detector,
  a drift chamber with 30 \unt{cm} outer radius,
  which is also important as part of the trigger system, 
  and a time projection chamber with 180 \unt{cm} outer radius.
  Calorimetry proceeds in two stages: electromagnetic and hadronic.
  A muon detection system of two double-layers of streamer tubes surrounds
  the whole detector. Finally, important for precise cross-section
  measurement are the highly segmented luminosity calorimeters.

  Presented here is a summary of the main features of the apparatus, 
  more detailed accounts can be found elsewhere \cite{ALEPH_DET}, \cite{NIM}. 

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\subsection{The Central Tracking Chambers}
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\subsubsection{The Vertex Detector (VDET)}

  Close to the interaction point, tracking is performed by a silicon vertex
  detector. Vertex detector hits are used to provide additional precision
  for tracks already reconstructed in the outer tracking.

  The device is formed from 96 silicon wafers each of dimensions 
  $(5.12\times 5.12\times 0.03)$ $cm$, arranged in two coaxial
  cylinders around the beam pipe. 
  The spatial resolution,
  for normally incident tracks is: $\sigma_{r\phi} \approx$ 12 $\mu$\unt{m},
  $\sigma_{z} \approx$ 10 $\mu$\unt{m}.

\subsubsection{The Inner Tracking Chamber (ITC)}

  The vertex detector is surrounded by a conventional cylindrical 
  multiwire drift chamber. It measures the $r\phi$ position of a track
  on eight concentric layers of hexagonal drift cells at radii between
  16 and 26 $cm$, with adjacent layers staggered by half a cell width.
  The resolution depends on the drift length in the cell, with an average 
  of 150 $\mu m$. The position of tracks along the beam direction
  ($z$) is determined by measuring the difference in arrival time of the
  signals at each end of the wires. This, however, only has a resolution 
  of a few centimetres.
  The ITC tracking information is used to improve tracking resolution, and
  to provide information for the first level trigger decision (made within
  2-3 $\mu s$ after a beam crossing).
 
\subsubsection{The Time Projection Chamber (TPC)}

  The TPC is ALEPH's principal tracking detector providing 21 
  3D space-point measurements for fully contained tracks
  at radii between 30 and 180 \unt{cm}.
  It consists of a gas
  volume (at atmospheric pressure) bounded by inner and outer field cages
  and two end-plates.
  An electric field is created between a central membrane and the two
  end-plates (dividing the TPC chamber into two), giving a drift length
  of 2.2 \unt{m} on each side. As charged particles pass through the 
  chamber they leave a trail of ionized gas.
  The electrons drift along the electric field lines towards one 
  of the end-plates where they induce ionization avalanches in  
  proportional wire chambers. These are detected as pulses on
  cathode pads.
  The $\phi$ coordinate is calculated by interpolating the signals 
  induced on cathode pads and the $r$ coordinate is given 
  by the radial position of the pads involved in the measurement. 
  The $z$ coordinate of a point on a track's trajectory is obtained 
  from the electron drift time and the drift velocity.

  The five helix parameters shown in Figure \ref{HELIX}
  are determined by fitting a helix to the pad coordinates within
  the first half turn of each track candidate.
%
\begin{figure}[htb]
%\vspace{-10mm}
\begin{center}
\epsfig{file=figures/helix.eps,height=7cm}
\end{center}
%\vspace{-10mm}
\caption[Helix parameters used in the TPC tracking algorithms]
{
\label{HELIX}
Helix parameters used in the TPC tracking
algorithms. For this case all parameters are positive.
}
\end{figure}
%
  A circle fit made in the $xy$ plane yields $\omega$ (the inverse radius
  of curvature), $d_0$ (the impact parameter in the $xy$ plane),
  and $\phi_0$ (the emission angle in the $xy$ plane).
  A straight line fit in the $s_{xy}$-$z$ plane yields
  $z_0$ (the $z$ coordinate at the closest approach to the $z$ axis)
  and tan$\lambda$ (the tangent of the dip angle).
  The circle fit parameters are used to calculate the values of $s_{xy}$.

  The pulse height information on the wires provides information
  about the energy lost by the particle as it traverses the TPC.
  The rate of energy loss ($dE/dx$) is dependent on the particle mass
  and so can be used to identify different particle species.
  Electron identification is good, with greater than $3 \sigma$
  separation up to $p \approx$ 8 \unt{GeV/c}.
  The $\pi$-$K$ separation is roughly constant above 
  $p \approx$ 2 \unt{GeV/c}
  at about $2\sigma$, while the $K$-$p$ separation is only about $1\sigma$.
  Therefore, kaon and proton identification can be accomplished only
  on a statistical basis; nonetheless, it is an important means of reducing
  combinatorial background in many analyses.
  The azimuthal
  coordinated resolution is $\sigma_{r\phi} =$ 180 $\mu$\unt{m}
  at $0^\circ$ pad crossing angle. The $z$ spatial resolution for wires is
  $\sigma_{z} =$ 1.2 \unt{mm} (with a small $z$ dependence),
  and 0.8 \unt{mm} for pads (at $\theta = 90^\circ$).

  The transverse momentum resolution and impact parameter
  resolution are shown in Table \ref{TAB_TPCRES} for the TPC only,
  for the TPC and drift chamber, and for all three tracking detectors
  together. At low momentum (less than 0.4 \unt{GeV}) a constant term of 
  0.5\% should be added to the resolution due to multiple scattering.
\begin{table}[b]
\vspace{5mm}
\begin{center}
\caption[Tracking momentum and impact parameter resolution.]
{
\label{TAB_TPCRES}
      Momentum and impact parameter resolution for the TPC, the TPC+ITC,
      and the TPC+ITC+VDET.
      At low momentum a constant term of 0.5\% should be
      added to the momentum resolution due to multiple scattering.
}

\begin{tabular}{|l|c|c|c|}
\hline
  & Transverse momentum              & \multicolumn{2}{|c|}{Impact parameter}\\
  & resolution ($p_T$ in \unt{GeV})  & \multicolumn{2}{|c|}{resolution      }\\
\hline
Detector & $\Delta p_T / p_T$ &
           $\sigma_{r\phi} (\mu m)$ &
           $\sigma_{rz   } (\mu m)$ \\
\hline
    TPC    & $1.2 \times 10^{-3}$ $p_T$ & 310 & 808 \\
  + ITC    & $0.8 \times 10^{-3}$ $p_T$ & 107 & 808 \\
  + VDET   & $0.6 \times 10^{-3}$ $p_T$ &  23 &  28 \\
\hline
\end{tabular}
\end{center}
\end{table}

\subsection{The Principal Calorimeters}


\subsubsection{The Electromagnetic Calorimeter (ECAL)}

  The ECAL is a sampling proportional wire calorimeter consisting
  of lead sheets and proportional wire chambers covering the angular range
  $|\cos \theta| <$ 0.98. The calorimeter stops and measures the energy
  of electrons and photons.
  The ECAL is formed from a barrel surrounding the TPC, closed at each end
  by an end-cap. These are divided into 12 modules, each covering
  an azimuthal angle of $30^\circ$.
  The cracks between the modules, where 
  the ECAL is not sensitive to particles, constitute 2\% of the barrel
  surface and 6\% of the endcap surface. To ensure that the cracks in the
  endcaps and the barrel are not coincident the endcap modules are rotated
  through $15^\circ$ azimuth. The modules have 45 lead/wire-chamber layers,
  with a total thickness of 22 radiation lengths.

  The energy and position of each shower is read out
  using small cathode pads with dimensions $\sim (30 \times 30)$ \unt{mm}
  arranged to form towers pointing to the interaction point;
  each tower is read out in three segments in depth, known as storeys,
  with thicknesses of 4, 9, and 9 radiation lengths respectively.
  There are 74,000 such towers, corresponding to an average granularity
  of $0.9^\circ \times 0.9^\circ$. This fine segmentation is important in
  the identification of photons, electrons and neutral pions.
  The angular resolution of the ECAL is:
\begin{equation}
  \sigma_{\theta,\phi} = \left( \frac{2.5}{\sqrt{E}} + 0.25 \right) mrad,
\end{equation}
  where $E$ is measured in \unt{GeV}.

    The energy resolution, determined by comparing the measured energy to the
    track momentum or beam energy, is shown in Figure \ref{FIG_ECAL_RES}
    {\it (this is an example of a missing reference, the figure labelled
    FIG\_ECAL\_RES does not exist, the symbols ?? are shown instead of a number)}
    as a function of the electron energy. The corresponding fitted resolution is:
\begin{equation}
  \frac{\sigma_E}{E}=\frac{ 0.18 }{ \sqrt{E(GeV)} } + 0.009
\end{equation}
    where a linear sum is used to as it gives the best fit; using a quadratic
    form would give:
\begin{equation}
  \frac{\sigma_E}{E}=\frac{ 0.178 }{ \sqrt{E(GeV)} } \oplus 0.019
\end{equation}
    The energy resolution as a function of polar angle is plotted in
    {\it and so on.....}

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