 %\documentstyle[amsfonts,epsfig,portland]{aipproc_old}
\documentstyle[epsfig,graphicx]{aipproc}
%-%\documentclass{aipproc}
%-%\usepackage{epsfig}
%-%\usepackage{graphicx}
%
\DeclareGraphicsExtensions{.eps,.ps,.eps.gz,.ps.gz}
\DeclareGraphicsRule{.ps.gz}{eps}{.ps.bb}{`gunzip -c #1}
%%\DeclareGraphicsRule{.ps}{ps}{.ps}{#1}
\DeclareGraphicsRule{.eps.gz}{eps}{.eps.bb}{`gunzip -c #1}
%
\begin{document}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqar}{\begin{eqnarray}}
\newcommand{\eeqar}{\end{eqnarray}}
\newcommand{\bfig}{\begin{figure}[htb]}
\newcommand{\efig}{\end{figure}}
\newcommand{\wink}{<\!\!\!\!\!\hbox{\rm )}\;}
\pagestyle{plain}
\pagenumbering{arabic}
\title{Physics of Particle Detection \footnote{ICFA Instrumentation
    School, National Accelerator Center, Faure, South Africa, March 26 -- April 6, 2001; http://alephwww.physik.uni-siegen.de/$\sim$grupen/capetown01/}}

\author{Claus Grupen}
\address{Department of Physics, University of Siegen \\ D-57068 Siegen,
Germany \\ e-mail: grupen@aleph.physik.uni-siegen.de }

\vspace*{-1cm}
\maketitle
\begin{abstract}
In this review the basic interaction mechanisms of charged and neutral
particles are presented. The ionization energy loss of
charged particles is fundamental to most particle detectors and is
therefore described in more detail. The production of electromagnetic
radiation in various spectral ranges leads to the detection of charged
particles in scintillation, Cherenkov and transition-radiation counters.
Photons are measured via the photoelectric effect, Compton scattering or
pair production, and neutrons through their nuclear interactions.

A combination of the various detector methods helps to identify
elementary particles and nuclei. At high energies absorption techniques  in
calorimeters provide additional particle identification and an accurate
energy measurement.
\end{abstract}

\section*{Introduction}
\noindent The detection and identification of elementary particles and nuclei is
of particular importance in high-energy, cosmic-ray, and nuclear
physics 
\cite{r:cgru:[1],r:cgru:[2],r:cgru:[3],Green,r:cgru:[4],r:cgru:[5],r:cgru:[6]}.  
Identification means that the mass of the particle and
its charge is determined. In elementary particle physics most
particles have unit charge. But in the study e.g.\ of the chemical
composition of primary cosmic rays or at heavy-ion colliders different 
charges must be distinguished.

%new
Every effect of particles or radiation can be used as a working
principle for a particle detector.
%endnew

The deflection of a charged particle in a magnetic field determines
its momentum $p$; the radius of curvature $\rho$ is given by 
\beq 
\rho \propto \frac{p}{z} = \frac{\gamma m_0 \beta c}{z}
\eeq 
where $z$ is the particle's charge, $m_0$ its rest mass and
$\beta= \frac{v}{c}$ its velocity. The particle velocity can be
determined e.g.\ by a time--of--flight method yielding 
\beq 
\beta \propto \frac{1}{\tau} \quad,
\eeq 
where $\tau$ is the flight time. A calorimetric measurement provides a 
determination of the
kinetic energy 
\beq 
E^{\rm kin} = (\gamma - 1) m_0 c^2
\eeq
where $\gamma = \frac{1}{\sqrt{1-\beta^2}}$ is the Lorentz factor.

From these measurements the ratio of $m_0/z$ can be inferred, i.e.\ 
for singly charged particles we have already identified the particle.
To determine the charge one needs another $z$-sensitive effect, e.g.\ 
the ionization energy loss 
\beq 
\frac{{\rm d}E}{{\rm d}x} \propto
\frac{z^2}{\beta^2} \ln (a \beta \gamma)
\eeq
($a$ is a material-dependent constant).

Now we know $m_0$ and $z$ separately. In this way even different
isotopes of elements can be distinguished.

The basic principle of particle detection is that every physics effect
can be used as an idea to build a detector. In the following we
distinguish between the interaction of charged and neutral particles.
In most cases the observed signature of a particle is its ionization,
where the liberated charge can be collected and amplified, or its
production of electromagnetic radiation which can be converted into a
detectable signal. In this sense neutral particles are only detected
indirectly, because they must first produce in some kind of
interaction a charged particle which is then measured in the usual
way.

%
\section*{Interaction of Charged Particles}
%

%
\subsection*{Kinematics}
%

%new
\noindent Four-momentum conservation allows to calculate the maximum energy
transfer of a particle of mass $m_0$ and velocity $v=\beta c$ to an
electron initially at rest to be \cite{r:cgru:[2]}
%endnew

\beq
E^{\rm max}_{\rm kin} = \frac{2 m_e c^2 \beta^2 \gamma^2}{1 + 2 \gamma
\frac{m_e}{m_0} + \left( \frac{m_e}{m_0} \right)^2} = \frac{2 m_e
p^2}{m_0^2 + m_e^2 + 2m_e E/c^2} \quad,
\label{E:cgru:5}
\eeq
here $\gamma = \frac{E}{m_0 c^2}$ is the Lorentz factor, $E$ the total
energy and $p$ the momentum of the particle. \\
\indent For low-energy particles heavier than the electron ($2 \gamma
\frac{m_e}{m_0} \ll 1$; $\frac{m_e}{m_0} \ll 1$) Eq.~(\ref{E:cgru:5}) 
reduces to
\beq
E^{\rm max}_{\rm kin} = 2 m_e c^2 \beta^2 \gamma^2 \quad.
\eeq
For relativistic particles ($E_{\rm kin} \approx E$; $pc \approx E$) one gets
\beq
E^{\rm max} = \frac{E^2}{E + m_0^2 c^2/2 m_e} \quad.
\label{E:cgru:7}
\eeq
For example, in a $\mu$--$e$ collision the maximum transferable energy is
\beq
E^{\rm max} = \frac{E^2}{E+11} \hspace{2cm} E \mbox{ in GeV}
\eeq
showing that in the extreme relativistic case the complete energy can be
transferred to the electron.

If $m_0 = m_e$, Eq.~(\ref{E:cgru:5})  is modified to
\beq
E^{\rm max}_{\rm kin} = \frac{p^2}{m_e + E/c^2} = \frac{E^2 - m_e^2 c^4}{E
+ m_e c^2} = E - m_e c^2 \quad.
\eeq
  
%
\subsection*{Scattering}
%
%
\subsubsection*{Rutherford Scattering}
%
\noindent The scattering of a  particle of charge $z$ on a target of nuclear charge
$Z$ is mediated by
the electromagnetic interaction (Fig.~\ref{F:cgru:3}).

\begin{figure}[ht!] % fig 3
\begin{center}
\includegraphics[height=5.6cm,width=0.5\textwidth,clip]{abb1}
\vspace{5pt}
\caption{Kinematics of Coulomb scattering of a particle of charge $z$ on a
target of charge $Z$}
\label{F:cgru:3}
\end{center}
\end{figure}

The Coulomb force between the incoming particle  and the target is written
as
\beq
\vec{F} = \frac{z \cdot e \cdot Z \cdot e}{r^2} \frac{\vec{r}}{r} \quad.
\eeq
For symmetry reasons the net momentum transfer is only perpendicular to
$\vec{p}$ along the impact parameter $b$
\beq
p_b = \int^{+\infty}_{-\infty} F_b {\rm d} t = \int^{+\infty}_{-\infty} \frac{z
\cdot Z \cdot e^2}{r^2} \cdot \frac{b}{r} \cdot \frac{{\rm d}x}{\beta c} \quad,
\eeq
with $b= r \sin \varphi$, d$t = {\rm d}x/v = {\rm d}x/\beta c$, and $F_b$
force perpendicular to $p$.
\beqar
p_b & = & \frac{z \cdot Z \cdot e^2}{ \beta c} \int^{+\infty}_{-\infty}
\frac{b\,
{\rm d}x}{(\sqrt{x^2 + b^2})^3} = \frac{z \cdot Z \cdot e^2}{\beta c b }
\underbrace{\int^{+\infty}_{-\infty} \frac{{\rm d} (x/b)}{\left(\sqrt{1 + 
\left(  
\frac{x}{b} \right)^2} \right)^3}}_{=2} \\
p_b & = & \frac{ 2z \cdot Z \cdot e^2}{\beta c b} = \frac{2 r_e m_e c}{b
\beta} z \cdot Z \quad,
\label{E:cgru:13}
\eeqar
where $r_e$ is the  classical electron radius.
This consideration leads to a scattering angle
\beq
\Theta = \frac{p_b}{p} = \frac{2 z \cdot Z \cdot e^2}{\beta c b} \cdot
\frac{1}{p} \quad.
\eeq
The cross section for this process is given by the well-known Rutherford
formula
\beq
\frac{{\rm d}\sigma}{{\rm d} \Omega} = 
\frac{z^2 Z^2 r^2_e}{4} 
\left( \frac{m_ec}{\beta p} \right)^2 
\frac{1}{\sin^4 \Theta/2} \quad.
\label{E:cgru:15}
\eeq
Figure \ref{F:scatteralphagold} shows the results of scattering $\alpha$ particles on gold foils \cite{Ruth,Geiger}.
\begin{figure}
\begin{center}
\includegraphics[height=8.75cm,width=0.65\textwidth,clip]{G1acompr}
\vspace{5pt}
\caption{Scattering of MeV $\alpha$ particles on gold foils}
\label{F:scatteralphagold}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
%\includegraphics[height=8.75cm,width=0.65\textwidth,clip]{Folie13}
\includegraphics[height=8.75cm,width=0.65\textwidth,clip]{folie13compr}
\vspace{5pt}
\caption{Scattering angle distribution for 15.7\,MeV electrons on two gold 
targets of different thickness \protect\cite{Hanson,Birkhoff} }
\label{F:scatterangleel}
\end{center}
\end{figure}

%
\subsubsection*{Multiple Scattering}
%
\noindent From Eq.~(\ref{E:cgru:15}) one can see that the average 
scattering angle $\langle
\Theta \rangle$ is zero. To characterize the different degrees of scattering
when a particle passes through an absorber one normally uses the
so-called ``average scattering angle'' $\sqrt{\langle \Theta^2 \rangle}$.
The projected angular distribution of scattering angles in this sense leads
to an average scattering angle of \cite{r:cgru:[6]}
\beq
\sqrt{\langle \Theta^2 \rangle} = \Theta_{\rm plane} = \frac{13.6\,{\rm
MeV}}{\beta cp} z \cdot \sqrt{\frac{x}{X_0}} \left\{ 1 + 0.038 \ln \left(
\frac{x}{X_0} \right) \right\}
\eeq
with $p$ in MeV/$c$ and $x$ the thickness of the scattering medium
measured
in radiation lengths $X_0$ (see {\bf bremsstrahlung}). The average
scattering angle
in three dimensions is
\beq
\Theta_{\rm space} = \sqrt{2}\, \Theta_{\rm plane} = \sqrt{2}\, \Theta_0 \quad.
\eeq
The projected angular distribution of scattering angles can  approximately
be represented by a Gaussian
\beq
P (\Theta) {\rm d} \Theta = \frac{1}{\sqrt{2 \pi} \Theta_0} \exp \left\{ -
\frac{\Theta^2}{2 \Theta^2_0} \right\} {\rm d}\Theta \quad.
\eeq
Figure~\ref{F:scatterangleel} shows the results of scattering 15.7\,MeV electrons 
off gold foils \cite{Hanson,Birkhoff}.
For low scattering angles ($\leq 5^{\circ}$) multiple scattering dominates. 
The distribution develops a tail for large scattering angles due to single 
scattering events.
%
\subsection*{Energy Loss of Charged Particles}
%

%new
\noindent Charged particles interact with a medium via electromagnetic
interactions by the exchange of photons.
If the range of photons is short, the absorption of virtual photons
constituting the field of the charged particle gives rise to ionization of
the material.
If the medium is transparent Cherenkov radiation can be emitted above
a certain threshold.
But also sub-threshold emission of electromagnetic radiation can occur, 
if discontinuities of the dielectric constant of the material are
present (transition radiation) \cite{r:cgru:[7]}.
The emission of real photons by decelerating a charged particle in a
Coulomb field also constitutes an important energy loss
(bremsstrahlung). 
%end new

%
\subsection*{Ionization Energy Loss}
%
%
\subsubsection*{Bethe--Bloch Formula}
%
\noindent This energy loss mechanism represents the scattering of charged
 particles off atomic electrons, e.g.
\beq
\mu^+ + \mbox{atom} \rightarrow \mu^+ + \mbox{atom}^+ + e^- \quad.
\label{E:cgru:22}
\eeq

The momentum transfer to the electron is (see Eq.~(\ref{E:cgru:13}))
\[
p_b = \frac{2 r_e m_e c}{b \beta} z \quad,
\]
and the energy transfer in the classical approximation
\beq
\varepsilon = \frac{p_b^2}{2 m_e} = \frac{2 r_e^2 m_e c^2}{b^2 \beta^2} z^2
\quad.
\label{E:cgru:23}
\eeq
The interaction probability per (g/cm$^2$), given the atomic cross section
$\sigma$, is
\beq
\phi [{\rm g}^{-1} {\rm cm}^2] = \frac{N}{A} \sigma\,[{\rm cm}^2/{\rm atom}]
\eeq
where $N$ is Avogadro's constant.

\begin{figure}[ht!] % fig
\begin{center}
\includegraphics[height=4.5cm,width=0.5\textwidth]{bild8compr}
\vspace{5pt}
\caption{Sketch explaining the differential collision probability}
\label{F:cgru:8}
\end{center}
\end{figure}

The differential probability to hit an electron in the area of an annulus with
radii $b$ and $b+{\rm d}b$
(see Fig.~\ref{F:cgru:8}) with an energy transfer between $\varepsilon$ and
$\varepsilon + {\rm d} \varepsilon$ is
\beq
\phi (\varepsilon) {\rm d} \varepsilon = \frac{N}{A} 2 \pi b\,{\rm d}b\,Z 
\quad,
\label{E:cgru:25}
\eeq
because there are $Z$ electrons per target atom.

Inserting $b$ from Eq.~(\ref{E:cgru:23}) into Eq.~(\ref{E:cgru:25}) gives
\beqar
b^2 & = & \frac{2 r_e^2 m_e c^2}{\beta^2} z^2 \cdot \frac{1}{\varepsilon}
\nonumber \\
2|b\,{\rm d}b| & = & \frac{2 r_e^2 m_e c^2}{\beta^2} z^2 \cdot \frac{{\rm
d}\varepsilon}{\varepsilon^2} \nonumber \\
\phi (\varepsilon) {\rm d} \varepsilon & = & \frac{N}{A} \pi \frac{2 r_e^2 m_e
c^2}{\beta^2} z^2 \cdot Z \cdot \frac{{\rm d}\varepsilon}{\varepsilon^2}
\nonumber \\
& = & \frac{2 \pi r_e^2 m_e c^2 N}{\beta^2} \cdot \frac{Z}{A} \cdot z^2 \cdot
\frac{{\rm d}\varepsilon}{\varepsilon^2} \quad,
\label{E:cgru:26}
\eeqar
showing that the energy spectrum of $\delta$ electrons or knock-on
electrons follows an $1/\varepsilon^2$ dependence (Fig.~\ref{F:cgru:9}, 
\cite{r:cgru:[8]}).

\begin{figure}[ht!] % fig
\begin{center}
\includegraphics[angle=0.2,height=7.5cm,width=0.55\textwidth,clip]{bild9compr}
\vspace{5pt}
\caption{$1/\varepsilon^2$ dependence of the knock-on electron
production probability \protect\cite{r:cgru:[8]} }
\label{F:cgru:9}
\end{center}
\end{figure}


The energy loss is now computed from Eq.~(\ref{E:cgru:25}) by integrating over 
all possible impact parameters \cite{r:cgru:[5]}
\beqar
- {\rm d} E & = & \int^{\infty}_0 \phi (\varepsilon) \cdot \varepsilon \cdot {\
d}x  \nonumber \\
& = & \int^{\infty}_0 \frac{N}{A} 2 \pi b \cdot {\rm d} b \cdot Z \cdot 
\varepsilon \cdot
{\rm d}x \nonumber \\
-\frac{{\rm d}E}{{\rm d}x} & = & \frac{2\pi N}{A} \cdot Z \int^{\infty}_0
\varepsilon \cdot b \cdot {\rm d} b \nonumber \\
& = & 2 \pi \frac{Z \cdot N}{A} \cdot \frac{2 r_e^2 m_e c^2}{\beta^2} z^2
\int^{\infty}_0 \frac{{\rm d} b}{b} \quad.
\label{E:cgru:27}
\eeqar
This classical calculation yields an integral which diverges for $b=0$ as
well as for $b=\infty$. This is not a surprise because one would not expect
that our approximations hold for these extremes.
\begin{itemize}
\item[a)] The $b=0$ case: Let us approximate the ``size'' of the
  target electron seen from the rest frame of the incident particle by
half the de Broglie wavelength. This gives a minimum impact parameter of
\beq
b_{\min} = \frac{h}{2 p} = \frac{h}{2 \gamma m_e \beta c} \quad.
\label{E:cgru:28}
\eeq
\item[b)] The $b=\infty$ case: If the revolution time $\tau_R$ of the electron
in the target atom becomes smaller than the interaction time $\tau_i$, the
incident particle ``sees'' a more or less neutral atom
\beq
\tau_i = \frac{b_{\rm max}}{v} \sqrt{1-\beta^2} \quad.
\eeq
\end{itemize}
The factor $\sqrt{1-\beta^2}$ takes into account that the field at high
velocities is Lorentz contracted. Hence the interaction time is shorter. For
the revolution time we have
\beq
\tau_R = \frac{1}{\nu_Z \cdot Z} = \frac{h}{I} \quad,
\eeq
where $I$ is the mean excitation energy of the target material, which
can be approximated by
\beq
I = 10 ~[\rm eV] \cdot Z
\eeq
for elements heavier than sulphur.
%end new

The condition to see the target as neutral now leads to
\[
\tau_R = \tau_i \qquad \Rightarrow \qquad \frac{b_{\rm max}}{v}
\sqrt{1-\beta^2} = \frac{h}{I}
\]
\beq
b_{\rm max} = \frac{\gamma h \beta c}{I} \quad.
\label{E:cgru:31}
\eeq
With the help of Eqs.~(\ref{E:cgru:28}) and (\ref{E:cgru:31}) we can solve 
the integral in Eq.~(\ref{E:cgru:27})
\beq
-\frac{{\rm d}E}{{\rm d}x} = 2 \pi \cdot \frac{Z}{A} N \cdot \frac{2 r_e^2 m_e
c^2}{\beta^2} z^2 \cdot \ln \frac{2 \gamma^2 \beta^2 m_e c^2}{I} \quad.
\eeq
Since for long-distance interactions the Coulomb field is screened by the
intervening  matter one has
\beq
-\frac{{\rm d}E}{{\rm d}x} = \kappa z^2 \cdot \frac{Z}{A} \frac{1}{\beta^2}
\left[ \ln \frac{2 \gamma^2 \beta^2 m_e c^2}{I} - \eta \right] \quad,
\label{E:cgru:33}
\eeq
where $\eta$ is a screening parameter (density parameter) and
\[
\kappa = 4 \pi N r^2_e m_e c^2 \quad.
\]
The exact treatment of the ionization energy loss of heavy particles leads
to \cite{r:cgru:[6]}
\beq
-\frac{{\rm d}E}{{\rm d}x} = \kappa z^2 \cdot \frac{Z}{A} \cdot
\frac{1}{\beta^2} \left[ \frac{1}{2} \ln \frac{2 m_e c^2 \gamma^2
\beta^2 E_{\rm kin}^{\rm max}}{I^2} - \beta^2 - \frac{\delta}{2} \right]
\eeq
which reduces to Eq.~(\ref{E:cgru:33}) for $\gamma m_e/m_0 \ll 1$ and $\beta^2 +
\frac{\delta}{2} = \eta$.

%new
The energy loss rate of muons in iron is shown in Fig.~
\ref{F:cgru:10}\cite{r:cgru:[6]}. It exhibits a
$\frac{1}{\beta^2}$ decrease until a minimum of ionization is obtained 
for $3 \leq \beta \gamma \leq 4$.
%end new

\begin{figure}[ht!] % fig
\begin{center}
\includegraphics[angle=0.2,width=0.7\textwidth,clip]
{fig4compr}
%\vspace{5pt}
\caption{Energy loss of muons in iron \protect\cite{r:cgru:[6]}}
\label{F:cgru:10}
\end{center}
\end{figure}

Due to the $\ln
\gamma$ term the energy loss increases again (relativistic rise, logarithmic
rise) until a plateau is reached (density effect, Fermi plateau).

The energy loss is usually expressed in terms of the area density
${\rm d}s = \rho {\rm d}x$
with $\rho$ -- density of the absorber.
It varies with the target material like $Z/A$ ($\leq 0.5$ for most
elements). Minimum-ionizing particles lose 1.94\,MeV/(g/cm$^2$) in
helium decreasing to 1.08\,MeV/(g/cm$^2$) in uranium. The energy loss
of minimum-ionizing particles in hydrogen is exceptionally large,
because here $Z/A = 1$.

The relativistic rise saturates at high energies because the medium
becomes polarized, effectively reducing the influence of distant
collisions. The density correction $\delta/2$ can be described by
\beq
\frac{\delta}{2} = \ln \frac{\hbar \omega_p}{I} + \ln \beta \gamma -
\frac{1}{2}
\label{E:cgru:new33}
\eeq
where
\beq
\hbar \omega_p = \sqrt{4 \pi N_e r^3_e} m_e c^2 / \alpha
\label{E:cgru:new34}
\eeq
is the plasma energy and $N_e$ the electron density of the absorbing
material. 

For gases the Fermi plateau, which saturates the relativistic rise, is
about $60\%$ higher compared to the minimum of ionization.
Figure~\ref{F:cgru:11} shows the measured energy loss rates of 
electrons, muons, pions, kaons, protons, and deuterons in the PEP4/9-TPC (185
d$E$/d$x$ measurements at 8.5\,atm in Ar-CH$_4$ = 80 : 20)
\cite{r:cgru:[newNM]}. Figure~\ref{F:alphaavalanche} shows the tracks of a 5\,MeV proton and a
19\,MeV $\alpha$ particle in an optical avalanche microdosimeter \cite{Titt}. 
The increase of the ionization towards the end of the range is due 
to the 1/$\beta^{2}$ dependence of the energy loss. The produced 
$\delta$ electrons show the same feature. The same behaviour is also seen in 
a double-gem (gas-electron multiplier) micro-strip gas chamber, where the 
scintillation is measured using a high-resolution CCD camera (Fig.~\ref{F:alphaCH4}, 
\cite{Fraga}).

The $z$ dependence of the ionization density can also clearly be seen from the
tracks of relativistic heavy ions in nuclear emulsions (Fig.~\ref{F:ionemulsion}, \cite{Rochester}). The 
strong enhancement of the ionization density at the end of the range, 
especially for heavy ions, can be used for cancer therapy (Fig.~\ref{F:Braggpeak}, \cite{Kraft}). 


\begin{figure}[ht!] % fig
\begin{center}
%\includegraphics[angle=0.2,width=0.7\textwidth,clip]
%{fig4}
%\end{center}
%%\vspace{5pt}
%\caption{Energy loss of muons in iron \protect\cite{r:cgru:[6]}}
%\label{F:cgru:10}
%\begin{center}
\includegraphics[width=0.7\textwidth,clip]{FIG5compr}
\vspace{5pt}
%\end{center}
\caption{Measured ionization energy loss of electrons, muons, pions, kaons,
protons and deuterons in the PEP4/9-TPC (Ar--CH$_4$ = 80 : 20 at 8.5\,atm) \protect\cite{r:cgru:[newNM]}}
\label{F:cgru:11}
\end{center}
\end{figure}

\begin{figure}[ht!] %
\begin{center}
\includegraphics[angle=-90,totalheight=7.0cm,width=1.\textwidth,clip]{out}
\vspace{5pt}
\caption{Tracks of a 5\,MeV proton and a 19\,MeV $\alpha$ particle in an optical
avalanche microdosimeter \protect\cite{Titt} }
\label{F:alphaavalanche}
\end{center}
\end{figure}

\begin{figure}[ht!] %
\begin{center}
\includegraphics[height=8.cm,width=0.8\textwidth,clip]{fig5bcompr}
\vspace{5pt}
\caption{Tracks of $\alpha$ particles in CF$_4$-based gas mixtures recorded
via the scintillation in a double-gem micro-strip gas chamber \protect\cite{Fraga}}
\label{F:alphaCH4}
\end{center}
\end{figure}

\begin{figure}[ht!] %height=12.cm
\begin{center}
\includegraphics[width=0.8\textwidth,clip]{fig5cnewcompr}
\vspace{5pt}
\caption{Tracks of relativistic heavy ions in nuclear emulsions \protect\cite{Rochester}}
\label{F:ionemulsion}
\end{center}
\end{figure}

\begin{figure}[ht!] %
\begin{center}
\includegraphics[height=8.cm,width=0.8\textwidth,clip]{folie21compr}
\vspace{5pt}
\caption{Ionization profiles of $^{12}$C ions in water for different beam
 energies. The strong ionization at the end of the range [Bragg peak] 
represents an ideal "scalpel" for the treatment of deep-seated tumours \protect\cite{Kraft}}
\label{F:Braggpeak}
\end{center}
\end{figure}

%
\subsubsection*{Landau Distributions}
%
\noindent The Bethe--Bloch formula describes the average energy loss of charged
particles. The fluctuation of the energy loss around the mean is described
by an asymmetric  distribution, the Landau distribution 
\cite{r:cgru:[10],r:cgru:[11]}.

The probability $\phi (\varepsilon) {\rm d}\varepsilon$ that a singly charged
particle loses an energy between $\varepsilon$ and $\varepsilon + {\rm
d}\varepsilon$ per unit length of an absorber was (Eq.~(\ref{E:cgru:26}))
\beq
\phi (\varepsilon) = \frac{2 \pi N e^4}{m_e v^2} \frac{Z}{A} \cdot
\frac{1}{\varepsilon^2} \quad.
\label{E:cgru:35}
\eeq
Let us define

\beq
\xi = \frac{2\pi Ne^4}{m_ev^2} \cdot \frac{Z}{A} x \quad,
%\label{E:cgru:35}
\eeq
where $x$ is the area density of the absorber:

\beq
\phi (\varepsilon) = \xi (x) \frac{1}{x \varepsilon^2} \quad.
\eeq
Numerically one can write

\beq
\xi = \frac{0.1536}{\beta^2} \frac{Z}{A} \cdot x \quad \mbox{[keV]} \quad,
\eeq
where $x$ is measured in mg/cm$^2$.

For an absorber of 1\,cm Ar we have for $\beta=1$
\[
\xi = 0.123\,{\rm [keV]} \quad.
\]
We define now
\beq
f(x, \Delta) = \frac{1}{\xi} \omega (\lambda)
\eeq
as the probability that the particle loses an energy $\Delta$ on traversing
an absorber of thickness $x$. $\lambda$ is defined to be the normalized
deviation from the most
probable energy loss $\Delta^{\rm m.p.}$
\beq
\lambda = \frac{\Delta - \Delta^{\rm m.p.}}{\xi} \quad.
\eeq
The most probable energy loss is calculated to be 
\cite{r:cgru:[10],r:cgru:[12]}
\beq
\Delta^{\rm m.p.} = \xi \left\{ \ln \frac{2 m_e c^2 \beta^2 \gamma^2 \xi}
{I^2} -
\beta^2 + 1- \gamma_{\rm E} \right\} \quad,
\eeq
where $\gamma_{\rm E} = 0.577 \ldots$ is Euler's constant.
The most probable energy loss in argon for minimum-ionizing particles is 
$\Delta^{\rm m.p.}= 1.2$\,{\rm keV}/cm while the average energy loss amounts
to 2.69\,{\rm keV}/cm.

Landau's treatment of $f(x, \Delta)$ yields
\beq
\omega (\lambda) = \frac{1}{\pi} \int^{\infty}_0 {\rm e}^{-u \ln u - \lambda u}
\sin \pi u \,{\rm d} u \quad,
\eeq
which can be approximated by \cite{r:cgru:[12]}
\beq
\Omega (\lambda) = \frac{1}{\sqrt{2 \pi}} \exp \left\{ - \frac{1}{2} (\lambda +
{\rm e}^{-\lambda}) \right\} \quad.
\eeq
Figure \ref{F:cgru:13} shows the energy-loss distribution of 3\,GeV 
electrons in an
Ar/CH$_4$ (80:20) filled drift chamber of 0.5\,cm thickness 
\protect\cite{r:cgru:[13]}.
According to Eq.~(\ref{E:cgru:35}) the $\delta$-ray contribution to the energy 
loss falls inversely proportional to the energy transfer squared, 
producing a long tail, called Landau tail, in the energy-loss distribution 
up to the kinematical limit (see also Fig.~\ref{F:cgru:9}).

\begin{figure}[!ht]
  \begin{center}
    \leavevmode
    \includegraphics[width=0.7\textwidth,clip]{landau_samb}
    \vspace{5pt}
    \caption{Energy loss distribution of 3\,GeV electrons in a thin-gap
      multi-wire drift chamber \protect\cite{r:cgru:[13]}}
    \label{F:cgru:13}
    \vspace{4.7mm}
  \end{center}
\end{figure}
\begin{figure}[ht!] %
\begin{center}
\includegraphics[width=1.\textwidth,clip]{paw}
\vspace{5pt}
\caption{Truncated energy-loss distributions for electrons, pions, kaons 
and protons in the ALEPH time projection chamber \protect\cite{bang}}
\label{F:alephdedx}
\end{center}
\end{figure} 

The asymmetric property of the energy-loss distribution becomes obvious
for thin absorbers. For larger
absorber thicknesses or truncation techniques applied to thin absorbers
the Landau  distribution  gets more symmetric. The truncated Landau 
distribution for electrons, pions, kaons, and protons in various momentum ranges
 are shown in Fig.~\ref{F:alephdedx} \cite{bang}.
%
\subsubsection*{Channeling}
%
\noindent The energy loss of charged particles as described by the Bethe--Bloch formula
needs to be modified for crystals where the collision partners are arranged
on a regular lattice. By looking into a crystal (Fig.~\ref{F:Viewcrystal}), it becomes 
immediately clear that the energy loss along certain crystal directions will 
be quite different from that along a non-aligned direction or in an amorphous
substance. The motion along such channeling directions is governed mainly by 
coherent scattering on strings and planes of atoms rather than by the 
individual scattering off single atoms. This leads to anomalous energy losses 
of charged particles in crystalline materials \cite{Moeller}.
\begin{figure}[ht!] %
\begin{center}
\includegraphics[height=8.cm,width=0.6\textwidth,clip]{fig6bcompr}
\vspace{5pt}
\caption{View into a crystal lattice}
\label{F:Viewcrystal}
\end{center}
\end{figure}


\begin{figure}[ht!] %
\begin{center}
\includegraphics[height=8.5cm,width=0.7\textwidth,clip]{fig6ccompr}
\vspace{5pt}
\caption{The energy-loss spectra for 15\,GeV/$c$ protons passing through a 740 $\mu$m thick germanium crystal
\protect\cite{Gemmel}}
\label{F:pGe}
\end{center}
\end{figure}
It is obvious from the crystal structure that charged particles can only be
channeled along a crystal direction if they are moving more or less parallel
 to the crystal axis. The critical angle necessary for channeling is small
 (approx. $1^{\circ}$ for $\beta \approx$ 0.1) and decreases with energy. For the 
axial direction $\langle111\rangle$, it can be estimated by 
\beq
\psi\,[\rm degrees] = 0.307 \cdot (z \cdot Z/E \cdot d)^{0.5}
\eeq
where $z$ and $Z$ are the charges of the incident particle and the crystal, $E$
 the particle's energy in MeV, and $d$ the interatomic spacing in $\rm \AA$. $\psi$ is 
measured in degrees \cite{Gemmel}.

For protons ($z$ = 1) passing through a silicon crystal ($Z=14$; $d=2.35$ $\rm \AA$) the
critical angle for channeling along the direction of body diagonals becomes
\beq
\psi = 13\,\mu \rm rad/\,\sqrt{\sl E\,[\rm TeV]} \quad .
\eeq

For planar channeling along the face diagonals ($\langle110\rangle$ axis) in 
silicon one gets
\beq
\psi = 5\,\mu \rm rad/ \sqrt{\sl E\,[\rm TeV]} \quad .
\eeq
Of course, the channeling process also depends on the charge of the incident
particle.

In a silicon crystal the positive nuclear charges produce an electric field 
of $2 \cdot 10^{12}$ V/cm at a distance of 0.1 $\rm \AA$ from an individual silicon 
nucleus. This field, however, decreases rapidly (like $1/r^{2}$) and therefore
 extends only over small distances. In contrast, for a string of silicon 
atoms along the $\langle110\rangle$ crystal direction one obtains a field of 
$1.3 \cdot 10^{10}$ V/cm. This field extends over macroscopic distances and 
can be used for the deflection of high-energy charged particles using bent 
crystals \cite{Gemmel}.

Channeled positive particles are kept away from a string of atoms and 
consequently suffer a relatively small energy loss. Figure~\ref{F:pGe} shows the 
energy-loss spectra for 15\,GeV/$c$ protons passing through a 740\,$\mu$m thick 
germanium crystal \cite{Gemmel}. The energy loss of channeled protons is lower by about 
a factor of two compared to random directions through the crystal.

%
\subsection*{Scintillation in Materials}
%
\noindent Scintillator materials can be inorganic  crystals, organic liquids or
plastics, 
and gases. The scintillation mechanism in organic crystals is an effect of
the
lattice. Incident particles can transfer energy to the lattice by creating
electron--hole pairs or taking electrons to higher energy levels below the
conduction band. Recombination of electron--hole pairs may lead to the
emission of light. Also electron--hole bound states (excitons) moving
through the lattice can emit light when hitting an activator center and
transferring their binding energy to activator levels, which subsequently
deexcite. In thallium-doped NaI crystals about 25\,eV are required to
produce one scintillation photon. The decay time in inorganic scintillators
can be quite long (1$\mu$s  in CsI(Tl); 0.62\,$\mu$s in BaF$_2$).

In organic substances the scintillation mechanism is different. Certain
types of molecules will release a small fraction ($\approx$ 3\%) of the
absorbed
energy as optical photons. This process is expecially marked in organic
substances which contain aromatic rings, such as polystyrene,
polyvinyltoluene, and naphtalene. Liquids which scintillate include toluene
or xylene \cite{r:cgru:[6]}.

This primary scintillation light is preferentially emitted in the
UV range. The absorption length for UV photons in the scintillation material
is rather short: the scintillator is not transparent for its own scintillation
light. Therefore, this light is transferred to a wavelength shifter which
absorbs the UV light and reemits it at longer wavelengths (e.g.\ in the
green). Due to the lower concentration of the wavelength-shifter material
the reemitted light can get out of the  scintillator and be detected by a
photosensitive device. The technique of wavelength shifting is also used
to match the emitted light to the spectral sensitivity of the photomultiplier.
For plastic scintillators the primary scintillator and wavelength shifter are
mixed with an organic material to form a polymerizing structure. In liquid
scintillators the two active components are mixed with an organic base 
\cite{r:cgru:[2]}.

About 100\,eV are required to produce one photon in an organic
scintillator. The decay time of the light signal in plastic scintillators is
substantially shorter compared to inorganic substances (e.g.\ 30\,ns in
naphtalene).

Because of the low light absorption in gases there is no need for
wavelength shifting in gas scintillators.

Plastic scintillators do not respond linearly to the energy-loss density. The
number of photons produced by charged particles is described by Birk's
semi-empirical formula \cite{r:cgru:[6],r:cgru:[17],r:cgru:[18]}
\beq
N = N_0 \frac{{\rm d}E/{\rm d}x}{1 + k_{\rm B}\,{\rm d}E/{\rm d}x} \quad,
\label{E:cgru:49}
\eeq
where $N_0$ is  the photon yield at low specific ionization density, and
$k_{\rm B}$ is Birk's density parameter. For 100\,MeV protons in plastic
scintillators one has d$E/$d$x \approx 10$\,MeV/(g/cm$^2$)  and $k_{\rm B}
\approx 5$\,mg/(cm$^2$MeV), yielding a saturation effect of $\approx 5$\% 
\cite{r:cgru:[4]}.

For low energy losses Eq.~(\ref{E:cgru:49}) leads to a linear dependence
\beq
N = N_0 \cdot {\rm d}E/{\rm d}x \quad,
\eeq
while for very high d$E/$d$x$ saturation occurs at
\beq
N = N_0/k_{\rm B} \quad.
\eeq
There exists a correlation between the energy loss of a particle that goes
into the creation of electron--ion pairs or the production of scintillation 
light because electron--ion pairs can recombine thus reducing the
d$E/$d$x|_{\rm ion}$ signal. On the other hand the scintillation-light signal
is enhanced because recombination frequently leads to excited states
which deexcite yielding scintillation light.

%
\subsection*{Cherenkov Radiation}
%
\noindent A charged particle traversing a medium with refractive index $n$ with a
velocity $v$ exceeding the velocity of light $c/n$ in that medium, emits
Cherenkov radiation. The threshold condition is given by 
%eq. \ref{E:cgru:22}
\beq
\beta_{\rm thres} = \frac{v_{\rm thres}}{c} \geq \frac{1}{n} \quad.
\eeq
The angle of emission increases with the velocity reaching a maximum
value for $\beta=1$, namely
\beq
\Theta_c^{\rm max} =  \arccos \frac{1}{n} \quad.
\eeq
The threshold velocity translates into a threshold energy
\beq
E_{\rm thres} = \gamma_{\rm thres} m_0 c^2
\eeq
yielding
\beq
\gamma_{\rm thres} = \frac{1}{\sqrt{1-\beta^2_{\rm thres}}} =
\frac{n}{\sqrt{n^2-1}} \quad.
\label{gammathreshold}
\eeq

\noindent
The number of Cherenkov photons emitted per unit path length d$x$ is
\beq
\frac{{\rm d}N}{{\rm d}x} = 2 \pi \alpha z^2 \int \left( 1 - 
\frac{1}{n^2\beta^2}\right) \frac{{\rm d}\lambda}{\lambda^2}
\label{E:cgru:56}
\eeq
for $n(\lambda) >1$, $z$ -- electric charge of the incident particle,
$\lambda$ -- wavelength, and $\alpha$ -- fine-structure constant. The  yield
of Cherenkov-radiation photons is proportional to $1/\lambda^2$, but only for 
those wavelengths where the refractive index is larger than unity. Since
$n(\lambda) \approx 1$ in the X-ray region, there is no X-ray Cherenkov
emission. Integrating Eq.~(\ref{E:cgru:56})  over the visible spectrum 
($\lambda_1 =
400$\,nm, $\lambda_2 = 700$\,nm) gives

\beqar
\frac{{\rm d}N}{{\rm d}x} & = & 2 \pi \alpha z^2 \frac{\lambda_2 -
\lambda_1}{\lambda_1 \lambda_2} \sin^2 \Theta_c \nonumber \\
& = & 490 \cdot z^2 \cdot \sin^2 \Theta_c \,[{\rm cm}^{-1}] \quad.
\eeqar

%new
The Cherenkov
effect can be used to identify particles of fixed momentum by means of
threshold Cherenkov counters. Figure~\ref{F:3.5aerogel} shows the pulse height distribution
for 3.5\,GeV/$c$ pions and protons in an aerogel Cherenkov counter \cite{BELLE}.
 For an index 
of refraction of $n=1.015$ the threshold Lorentz factor for Cherenkov 
radiation is, according to Eq.~(\ref{gammathreshold}), $\gamma=5.84$. This threshold is exceeded 
for pions ($\gamma_{\pi}=24.2$) but not for protons ($\gamma_{p}=2.86$). 
Therefore protons deposit energy only due to ionization while pions produce
 in addition Cherenkov light.
More information can be obtained, if the Cherenkov angle is measured
by DIRC counters (Detection of Internally Reflected Cherenkov
light). In these devices some fraction of the Cherenkov light produced 
by a charged particle is kept inside the radiator by total internal
reflection. The direction of the photons remains unchanged and the
Cherenkov angle is conserved during the transport. When exiting the
radiator the photons produce a Cherenkov ring on a planar detector
 (Fig.~\ref{F:cgru:new7}).

\begin{figure}[ht!] %
\begin{center}
%\includegraphics[height=8.cm,width=0.6\textwidth,clip]{fig6dn}
\includegraphics[height=8.cm,width=0.6\textwidth,clip]{fig6dcompr}
\vspace{5pt}
\caption{Pulse-height distribution for 3.5\,GeV/$c$ pions and protons in an 
aerogel Cherenkov counter \protect\cite{BELLE}}
\label{F:3.5aerogel}
\end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.7\textwidth,clip]{fig7compr}
    \vspace*{0.5cm}
    \caption{Imaging principle of a DIRC counter
      \protect\cite{r:cgru:Adam}}
    \label{F:cgru:new7}
  \end{center}
\end{figure}

The pion/proton separation achieved with such a system is shown in
Fig.~\ref{F:cgru:new8} \cite{r:cgru:Adam}.

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.7\textwidth,clip]{fig8compr}
    \caption{Cherenkov-angle distribution for pions and protons of 5.4 
      \,GeV/$c$ in a DIRC counter \protect\cite{r:cgru:Adam}.}
    \label{F:cgru:new8}
  \end{center}
\end{figure}

Ring-imaging Cherenkov counters (RICH counters) have become
extraordinary useful in the field of elementary particles and
astrophysics. Figure \ref{F:cgru:new9} shows the Cherenkov-ring radii of
electrons, muons, pions, and kaons in a C$_4$F$_{10}$--Ar (75:25)-filled
Rich counter read out by a 100-channel photomultiplier of $10\times10$
cm$^2$ active area. The resulting $\pi/\mu/e$ separation
for 3\,GeV/$c$ particles is shown in Fig.~\ref{F:cerenkovradiinew} \cite{r:cgru:Debbe}.


\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.65\textwidth,clip]{fig9compr}
    \caption{Cherenkov-ring radii of $e, \mu, \pi, K$ in a
      C$_4$F$_{10}$-Ar (75:25) RICH counter. The solid curves show the
      expected radii for an index of refraction of $n = 1.00113$. The
      shaded regions represent a 5\% uncertainty in the absolute
      momentum scale \protect\cite{r:cgru:Debbe}}
    \label{F:cgru:new9}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.65\textwidth,clip]{fig9bcompr}
    \caption{Cherenkov-ring radii for pions, muons, and electrons
     of 3\,GeV in a C$_4$F$_{10}$-Ar (75:25) RICH counter 
     \protect\cite{r:cgru:Debbe}.}
    \label{F:cerenkovradiinew}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.65\textwidth,clip]{tscan_pic_01}
    \caption{Neutrino-induced muon with subsequent muon decay in 
    the Super\discretionary{-}{}{-}Kamiokande experiment \protect\cite{skamio} }
    \label{F:neutrinokamio}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.7\textwidth,clip]{containedmuon-crop}
    \caption{Neutrino-induced muon in the SNO experiment \protect\cite{sno}.}
    \label{F:9d}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.7\textwidth,clip]{decayelectron}
    \caption{Cherenkov-ring produced by an electron from muon decay, where the muon was created by a 
    muon neutrino \protect\cite{sno}.}
    \label{F:9e}
  \end{center}
\end{figure}

An important aspect of neutrino physics with atmospheric neutrinos is 
the correct identification of neutrino-induced muons and electrons. A deficit
of neutrino-induced muons would hint at neutrino oscillations, where the 
$\nu_{\mu}$ neutrinos born in the decay of pions
($\pi^{+} \to \mu^{+} + \nu_{\mu}$) or muons ($\mu^{-} \to e^{-} + \bar{\nu_{e}} + \nu_{\mu}$) could have 
transformed themselves into $\tau$ neutrinos ($\nu_{\tau}$) or some sterile
neutrino (similarly for $\bar{\nu_{\mu}}$). Figure~\ref{F:neutrinokamio} shows a neutrino-induced
 event ($\nu_{\mu} + N \to \mu^{-} + X$) with subsequent 
decay of the $\mu^{-}$($\mu^{-} \to e^{-} + \bar{\nu_{e}} + \nu_{\mu}$) in 
the Super-Kamiokande experiment \cite{skamio}. Both rings due to the $\mu^{-}$ and 
$e^{-}$ are seen. Figures~\ref{F:9d} and \ref{F:9e} show a similar event in the SNO experiment \cite{sno}.
 In the first frame the muon ring is visible (event time 22.1867771 sec) and
 0.9 $\mu$s later the electron ring. These events clearly demonstrate the
particle identification capability of large-volume neutrino detectors. 
%end new
\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.6\textwidth,clip]{fig9fcompr}
    \caption{Electron/muon separation in the NOMAD transition-radiation detector 
    \protect\cite{normad}}
    \label{F:NOMAD}
  \end{center}
\end{figure}
\newpage
%
\subsection*{Transition Radiation}
%
\noindent Transition radiation is emitted when a charged particle traverses a medium
with discontinuous dielectric  constant. A charged particle moving  towards
a boundary, where the dielectric constant changes, can be considered to
form together with
its mirror  charge an electric  dipole whose field strength varies in time. 
The
time-dependent dipole field causes the emission of electromagnetic
radiation. This emission can be understood in such a way that although the
dielectric displacement $\vec{D} = \varepsilon \varepsilon_0 \vec{E}$ varies
continuously in passing through a boundary, the electric field does not.

The energy radiated from a single boundary (transition from vacuum to a
medium with dielectric constant $\varepsilon$) is proportional to the
Lorentz factor of the incident charged particle 
\cite{r:cgru:[6],r:cgru:[17],r:cgru:[21]}:
\beq
S = \frac{1}{3} \alpha z^2 \hbar \omega_p \gamma \quad,
\eeq
where
$\hbar \omega_p$ is the plasma energy (see Eq.~(\ref{E:cgru:new34})). 
For commonly used plastic radiators (styrene or similar
materials) one has

\beq
\hbar \omega_p \approx 20\,{\rm eV} \quad.
\eeq

The typical emission angle of transition radiation is proportional to
$1/\gamma$. The radiation yield drops sharply for frequencies
\beq
\omega > \gamma \omega_p \quad.
\eeq
The $\gamma$ dependence of the emitted energy originates mainly from
the hardening of the spectrum rather than from the increased photon
yield. Since the radiated photons also have energies proportional to the
Lorentz
factor of the incident particle, the number of emitted transition-radiation
photons is
\beq
N \propto \alpha z^2 \quad.
\eeq
The number of emitted photons can be increased by using many transitions
(stack of foils, or foam). 
%new
At each interface the emission probability for an X-ray photon is of
the order of $\alpha = 1/137$.
%end new
However, the foils or foams have to be of low-$Z$ material to avoid absorption in the radiator. 
Interference effects for
radiation from transitions in periodic arrangements  cause an effective
threshold behaviour at a value of $\gamma \approx 1000$. These effects
also produce a frequency-dependent photon yield. The foil thickness must
be comparable to or larger than the formation zone
\beq
D = \gamma c/\omega_p
\eeq
which in practical situations ($\hbar \omega_p = 20\,$eV; $\gamma = 5
\cdot 10^3$) is about 50\,$\mu$m.
%new
Transition radiation detectors are mainly used for
$e/\pi$ separation. In cosmic-ray experiments transition-radiation
emission can also be employed to measure the energy of muons in the
TeV range. Figure~\ref{F:NOMAD} \cite{normad} shows the muon/electron-discrimination
 power for 10\,GeV particles in the NOMAD transition-radiation detector. Muons
fall below the TRD threshold and deposit energy loss due to ionization only, 
while electrons of 10\,GeV ($\gamma_{\rm e} = 20\,000$) clearly produce transition radiation. 

%end new

%
\subsection*{Bremsstrahlung}
%
\noindent If a charged particle is decelerated in the Coulomb field of a nucleus a
fraction of its kinetic energy will be emitted in form of real photons
 (bremsstrahlung). The energy loss by bremsstrahlung for high energies 
can be described by \cite{r:cgru:[2]}
\beq
-\frac{{\rm d}E}{{\rm d}x} = 4 \alpha N_{\rm A} \frac{Z^2}{A} \cdot z^2 r^2 E \ln
\frac{183}{Z^{1/3}} \quad,
\label{E:cgru:65}
\eeq
where $r= \frac{1}{4\pi \varepsilon_0} \cdot \frac{e^2}{m c^2}$.
Bremsstrahlung is mainly produced by electrons because
\beq
r_e \propto \frac{1}{m_e} \quad.
\eeq
Equation~(\ref{E:cgru:65}) can be rewritten for electrons
\beq
-\frac{{\rm d}E}{{\rm d}x} = \frac{E}{X_0} \quad,
\eeq
where
\beq
X_0 = \frac{A}{4 \alpha N_{\rm A} Z (Z+1) r_e^2 \ln (183\, Z^{-1/3})}
\label{E:cgru:68}
\eeq
\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.6\textwidth,clip]{cosmocompr}
    \caption{Muon bremsstrahlung event in the ALEPH detector 
    \protect\cite{aleph}}
    \label{F:ALEPHmuonbremss}
  \end{center}
\end{figure}
\noindent is the radiation length of the absorber in which bremsstrahlung is
produced. Here we have included also radiation from electrons  ($\propto Z$,
because there are $Z$ electrons per nucleus). If screening effects are
taken into account $X_0$ can be more accurately described by 
\cite{r:cgru:[6]}
\beq
X_0 = \frac{716.4\,A}{Z (Z+1) \ln (287/\sqrt{Z})} \quad [{\rm g/cm}^2] \quad.
\eeq
Figure\,\ref{F:ALEPHmuonbremss} \cite{aleph} shows a bremsstrahlung event of a cosmic-ray muon in the 
electromagnetic calorimeter of the ALEPH experiment. The produced photon 
initiates an electromagnetic shower and the electrons and positrons in the 
cascade are deflected in opposite directions in the transverse magnetic field in the time projection
 chamber.
 
 
The important point about bremsstrahlung is that the energy loss is
proportional to the energy. The energy where the losses due to ionization
and bremsstrahlung for electrons are the same is called critical energy
\beq
\left. \frac{{\rm d}E_{\rm c}}{{\rm d}x} \right|_{\rm ion} = \left. \frac{{\rm
d} E_{\rm c}}{{\rm d}x} \right|_{\rm brems} \quad.
\eeq
For solid or liquid absorbers the critical energy can be approximated by 
\cite{r:cgru:[6]}
\beq
 E_{\rm c} = \frac{610\,{\rm MeV}}{Z + 1.24} \quad,
\eeq
while for gases one has \cite{r:cgru:[6]}
\beq
 E_{\rm c} = \frac{710\,{\rm MeV}}{Z + 0.92} \quad.
\eeq
The difference between gases on the one hand and solids and liquids on
the other hand comes about because the density corrections are different
in these substances, and this modifies $\left. \frac{{\rm d}E}{{\rm d}x}
\right|_{\rm ion}$.

The energy spectrum of bremsstrahlung photons is $\propto E_{\gamma}^{-1}$,
where $E_{\gamma}$ is the photon energy.

At high energies also radiation
from heavier particles becomes important  and consequently a critical energy
for these particles can be defined. Since
\beq
\left. \frac{{\rm d}E}{{\rm d}x} \right|_{\rm brems}  \propto
\frac{1}{m^2}
\eeq
the critical energy e.g.\ for muons in iron is
\beq
 E_{\rm c} = \frac{610\,{\rm MeV}}{Z + 1.24} \cdot \left( \frac{m_{\mu}}{m_e}
\right)^2 = 960\,{\rm GeV} \quad.
\eeq

%
\subsection*{Direct Electron Pair Production}
%
\noindent Direct electron pair production in  the Coulomb field of a nucleus
via virtual photons (``tridents'')  is a dominant energy-loss mechanism at
high energies. The energy loss for singly charged particles due to this
process
can be represented by

\beq
\left. -\frac{{\rm d}E}{{\rm d}x} \right|_{\rm pair}  = b (Z, A, E) \cdot E 
\quad.
\eeq

It is essentially -- like bremsstrahlung -- also proportional to the 
particle's energy. Because
bremsstrahlung and direct pair production dominate at high energies this
offers an attractive possibility to build also muon calorimeters 
\cite{r:cgru:[2]}.
The average rate of muon energy losses can be parametrized as

\beq
\frac{{\rm d}E}{{\rm d}x} = a(E) + b(E) \cdot E
\eeq
where $a(E)$ represents the ionization energy loss and $b(E)$ is the sum of
direct electron pair production, bremsstrahlung, and photonuclear 
interactions. 

%new
The various contributions to the energy loss of muons in standard rock
(Z = 11; A = 22; $\rho = 3\,\rm g/cm^3$) are shown in Fig.~\ref{F:cgru:new10}. 
\begin{figure}[ht!] % fig
\begin{center}
\includegraphics[width=0.9\textwidth,clip]
%{bild24}
{muoninrockcompr}
\caption{Contributions to the energy loss of muons in standard rock (Z
  = 11; A = 22; $\rho = 3\,\rm g/cm^3$).}
\label{F:cgru:new10}
\end{center}
\end{figure}

%end new

%
\subsection*{Nuclear Interactions}
%
\noindent Nuclear interactions play an important role in the detection of  neutral
particles other than photons. They are also responsible for the
development of hadronic cascades. The total cross section for nucleons is
of the order of 50\,mbarn and varies slightly with energy. It
has an elastic ($\sigma_{\rm el}$) and inelastic part ($\sigma_{\rm inel}$).
The inelastic cross section has a material dependence
\beq
\sigma_{\rm inel} \approx \sigma_0 A^{\alpha}
\eeq
with $\alpha = 0.71$. The corresponding absorption length $\lambda_a$ is
\cite{r:cgru:[2]}
\beq
\lambda_a = \frac{A}{N_{\rm A} \cdot \rho \cdot \sigma_{\rm inel}}\, [{\rm cm}]
\eeq
($A$ in g/mol, $N_{\rm A}$ in mol$^{-1}$, $\rho$ in g/cm$^3$, and $\sigma_{\rm
inel}$ in cm$^2$). \\
\indent This quantity has to be distinguished from the nuclear interaction
length $\lambda_w$, which is related to the total cross section
\beq
\lambda_w = \frac{A}{N_{\rm A} \cdot \rho \cdot \sigma_{\rm total}}\, [{\rm cm}]
\quad
\eeq
Since $\sigma_{\rm total} > \sigma_{\rm inel}$, $\lambda_w <
\lambda_a$ holds.

The total and elastic part of the proton--proton cross section are shown 
in Fig.~\ref{F:cgru:new10a} \cite{r:cgru:[6]} as a function of the laboratory energy $E_{\rm lab}$ and the 
center--of--mass energy $\sqrt{s}$, where $s = 2 \cdot m \cdot  E_{\rm lab}$
 (for $E_{\rm lab} \gg m_{\rm proton}$). 

Strong interactions have a multiplicity which grows  logarithmically with
energy. The particles  are produced  in a narrow cone around the forward
direction with an average transverse momentum of $p_{\rm T} = 350\,$MeV/$c$,
which is responsible for the lateral spread of hadronic cascades.

A useful relation for the calculation of interaction rates per (g/cm$^2$) is
\beq
\phi (({\rm g/cm}^2)^{-1}) = \sigma_{\rm N} \cdot N_{\rm A}
\eeq
where $\sigma_{\rm N}$ is the cross section per nucleon and $N_{\rm A}$
Avogadro's number.
\begin{figure}[ht!] % fig
\begin{center}
\includegraphics[width=1.\textwidth,clip]{hadronicrpp_page14}
\caption{The proton--proton cross section as a function of the laboratory energy 
and center--of--mass energy \protect\cite{r:cgru:[6]}.}
\label{F:cgru:new10a}
\end{center}
\end{figure}
%
\section*{Interaction of Photons}
%
\noindent Photons are attenuated in matter via the processes of the photoelectric
effect, Compton scattering, and pair production. The intensity of a photon
beam varies in matter according to
\beq
I = I_0 \:{\rm e}^{-\mu x} \quad,
\eeq
where $\mu$ is mass attenuation coefficient. $\mu$ is related to the
photon cross sections $\sigma_i$ by
\beq
\mu = \frac{N_{\rm A}}{A} \sum^3_{i=1} \sigma_i \quad.
\eeq

\subsection*{Photoelectric Effect}
\noindent Atomic electrons can absorb the energy of a photon completely
\beq
\gamma + \mbox{atom} \rightarrow \mbox{atom}^+ + e^- \quad.
\eeq
The cross section for absorption of a photon of energy $E_{\gamma}$ is
particularly large in the $K$ shell (80\% of the total cross section). The
total cross section for photon absorption in the $K$ shell is
\beq
\sigma^K_{\rm Photo} = \left( \frac{32}{\varepsilon^7} \right)^{1/2} \alpha^4
Z^5 \sigma_{\rm Thomson}\,[{\rm cm}^2/{\rm atom}] \quad,
\eeq
where $\varepsilon = E_{\gamma}/m_ec^2$, and $\sigma_{\rm Thomson} =
\frac{8}{3} \pi r_e^2 = 665$\,mbarn is the cross section for Thomson
scattering. For high energies the energy dependence becomes softer
\beq
\sigma^K_{\rm Photo} = 4 \pi r_e^2 Z^5 \alpha^4 \cdot \frac{1}{\varepsilon}
\quad.
\eeq
The photoelectric cross section has sharp discontinuities when
$E_{\gamma}$ coincides with the binding energy of atomic shells. As a
consequence of a photoabsorption in the $K$ shell characteristic X rays
or Auger electrons are emitted \cite{r:cgru:[2]}.

\subsection*{Compton Scattering}
\noindent The Compton effect describes the scattering of photons off  quasi-free
atomic electrons
\beq
\gamma + e \rightarrow \gamma' + e' \quad.
\eeq
The cross section for this process, given by the Klein--Nishina formula, can
be approximated at high energies by
\beq
\sigma_c \propto \frac{\ln \varepsilon}{\varepsilon} \cdot Z
\eeq
where $Z$ is the number of electrons in the target atom. From energy and
momentum conservation one can derive the ratio of scattered
($E_{\gamma}'$) to incident photon energy ($E_{\gamma}$)
\beq
\frac{E_{\gamma}'}{E_{\gamma}} = \frac{1}{1+\varepsilon(1-\cos
\Theta_{\gamma})} \quad,
\eeq
where $\Theta_{\gamma}$ is the scattering angle of the photon with
respect to its original direction.

For backscattering ($\Theta_{\gamma} = \pi$) the energy transfer to the
electron $E_{\rm kin}$ reaches a maximum value
\beq
E^{\rm max}_{\rm kin} = \frac{2 \varepsilon^2}{1+2 \varepsilon} m_ec^2 \quad,
\label{ekinmax}
\eeq
which, in the extreme case ($\varepsilon \gg 1$), equals $E_{\gamma}$.

In Compton scattering only a fraction of the photon energy is transferred to
the electron. Therefore, one defines an energy scattering cross section
\beq
\sigma_{cs} =  \frac{E_{\gamma}'}{E_{\gamma}} \sigma_c
\eeq
and an energy absorption cross section
\beq
\sigma_{ca} = \sigma_c - \sigma_{cs} = \sigma_c \frac{E_{\rm
kin}}{E_{\gamma}} \quad.
\eeq
At accelerators and in astrophysics also the process of inverse   Compton
scattering is of importance \protect\cite{r:cgru:[2]}.

\subsection*{Pair Production}
\noindent The production of an electron--positron pair in the Coulomb field of a
nucleus requires a certain minimum energy
\beq
E_{\gamma} \geq 2 m_ec^2 + \frac{2 m_e^2c^2}{m_{\rm nucleus}} \quad.
\eeq
Since for all practical cases $m_{\rm nucleus} \gg m_e$, one has effectively
$E_{\gamma} \geq 2 m_e c^2$.

The total cross section in the case of complete screening $\left( \varepsilon
\gg \frac{1}{\alpha Z^{1/3}} \right)$; i.e.\ at reasonably high energies
$(E_{\gamma} \gg 20\,$MeV), is
\beq
\sigma_{\rm pair} = 4 \alpha r^2_e Z^2 \left( \frac{7}{9} \ln
\frac{183}{Z^{1/3}} - \frac{1}{54} \right)\quad [{\rm cm}^2/{\rm atom}] \quad.
\label{E:cgru:104}
\eeq
Neglecting the small additive term $1/54$ in Eq.~(\ref{E:cgru:104}) one can 
rewrite,
using Eq.~(\ref{E:cgru:65}) and Eq.~(\ref{E:cgru:68}),
\beq
\sigma_{\rm pair} = \frac{7}{9} \frac{A}{N_{\rm A}} \cdot \frac{1}{X_0} \quad.
\eeq
The partition of the energy to the electron and positron is symmetric at low
energies $(E_{\gamma} \ll 50\,$MeV) and increasingly asymmetric at high
energies $(E_{\gamma} > 1\,$GeV) \protect\cite{r:cgru:[2]}.

Figure \ref{F:cgru:pairprod} shows the photoproduction  of an
electron--positron pair in the Coulomb field of an electron ($\gamma +
e^- \rightarrow e^+ + e^- + e^-$) and also a pair production in the
field of a nucleus ($ \gamma + \mbox{nucleus} \rightarrow e^+ + e^- +
\mbox{nucleus}^\prime$) \cite{r:cgru:Close}.

\begin{figure}[ht!]
  \begin{center}
 \includegraphics[angle=90,width=0.6\textwidth]{n_bild40}
\vspace*{0.5cm}
    \caption{Photoproduction in the Coulomb field of an electron 
($\gamma + e^- \rightarrow e^+ + e^- + e^-$) and on a nucleus 
($ \gamma + \mbox{nucleus} \rightarrow e^+ + e^- +
\mbox{nucleus}^\prime$)
\protect\cite{r:cgru:Close}}
    \label{F:cgru:pairprod}
  \end{center}
\end{figure}

\newpage
\subsection*{Mass Attenuation Coefficients}
\begin{figure}[bh!]
\begin{center}
%\begin{tabular}
\begin{minipage}[b]{.45\textwidth}
  \begin{center}
    \leavevmode
\includegraphics[angle=0.2,totalheight=7.0cm,width=\textwidth]{bild34compr}
\vspace{6pt}
\caption{Mass attenuation coefficients for photon interactions in silicon 
\protect\cite{r:cgru:[30]}}
\label{F:cgru:34}
  \end{center}
\end{minipage}
\hspace*{.04\textwidth}
\begin{minipage}[b]{.45\textwidth}
  \begin{center}
    \leavevmode
\includegraphics[totalheight=7.0cm,width=\textwidth,clip]
{bild35compr}
\vspace{6pt}
\caption{Mass attenuation coefficients for photon interactions in
germanium \protect\cite{r:cgru:[30]}}
\label{F:cgru:35}
%    \vspace{3.8mm}
  \end{center}
\end{minipage}
\vspace*{1.5cm}
\begin{minipage}[b]{.45\textwidth}
  \begin{center}
    \leavevmode
\includegraphics[angle=0.2,totalheight=7.0cm,width=\textwidth,clip]
{bild36compr}
\vspace{6pt}
\caption{Mass attenuation coefficients for photon interactions in lead 
\protect\cite{r:cgru:[30]}}
\label{F:cgru:36}
%    \vspace{4.4mm}
  \end{center}
\end{minipage}
\hspace*{.04\textwidth}
\begin{minipage}[b]{.45\textwidth}
%  \begin{center}
%    \leavevmode
\noindent The mass attenuation coefficients for photon interactions are shown in
Figs.~\ref{F:cgru:34}--\ref{F:cgru:36} for silicon, 
germanium, and lead \cite{r:cgru:[30]}. The photoelectric effect
dominates at low energies ($E_{\gamma} < 100\,$keV). Superimposed on the 
continuous photoelectric attenuation coefficient are absorption edges
characteristic of the absorber material. Pair production
dominates at high energies ($>10$\,MeV). In the intermediate region
Compton scattering prevails.

    \vspace{1.1cm}
%  \end{center}
\end{minipage}
%\end{tabular}


\end{center}
\end{figure}
A typical experiment set-up for the measurement of $\gamma$-ray spectra is
shown in Fig.~\ref{F:cgru:14a} \cite{tennessee}. Photons from a radioactive source enter a NaI(Tl)
 scintillation crystal and deposit energy either by the photoelectric effect, 
Compton scattering, or -- depending on the photon energy -- pair production. The 
amount of light produced by the electron is recorded in a photomultiplier.
A typical $\gamma$-ray spectrum of the 662\,keV $\gamma$ line from the decay
 of $^{137}$Cs into $^{137}$Ba is shown in Fig.~\ref{F:14b} \cite{Amptek}.
  Clearly 
visible is the photopeak (full absorption peak) at $E = 662$\,keV. Compton
scattering produces a continuum with a maximum energy transfer to electrons at
 478\,keV (see Eq.~(\ref{ekinmax})), the Compton edge. Photons which are backscattered
 from shielding material into the detector may undergo photoelectric 
effect thereby producing the backscatter peak at $E - E^{\rm max}_{\rm kin} = 184$\,keV.
 At low energies characteristic $X$-ray lines are visible originating from
the source or the shielding material. The spectra of $\gamma$-rays from a 
$^{60}$Co source recorded in a NaI(Tl) detector \cite{Amptek} and in a high-purity 
germanium counter \cite{hausmann} are compared in Fig.~\ref{F:cgru:new14c} and 
\ref{F:cgru:14d}. All features of 
$\gamma$-ray spectra like photopeaks at 1.17\,MeV and 1.33\,MeV, Compton edge,
backscatter peaks, and characteristic $X$ rays are clearly visible. Also the 
outstanding performance of the high-purity Ge counter with a rms resolution
of 400\,eV shows the superiority of solid-state detectors over scintillation 
counters.
 
\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=1.\textwidth,clip]{fig14acompr}
    \caption{Typical set-up for the measurement of $\gamma$-ray spectra \protect\cite{tennessee}}
    \label{F:cgru:14a}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.75\textwidth,clip]{fig14b}
    \caption{$\gamma$-ray spectrum of 662\,keV photons in a NaI(Tl) 
scintillation counter \protect\cite{Amptek}}
    \label{F:14b}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.75\textwidth,clip]{fig14c}
    \caption{$\gamma$-ray spectrum of $^{60}$Co measured in a NaI(Tl) 
scintillation detector \protect\cite{Amptek}}
    \label{F:cgru:new14c}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[angle=-90,width=0.8\textwidth]{co60}
    \caption{$\gamma$-ray spectrum of $^{60}$Co measured in a high-purity Ge detector 
    \protect\cite{hausmann}}
    \label{F:cgru:14d}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.6\textwidth,clip]{fig14ecompr}
    \caption{Spectra of characteristic X ray measured in a superconducting
 phase-transition thermometer \protect\cite{CRESST}}
    \label{F:cgru:14e}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.6\textwidth,clip]{fig14fcompr}
    \caption{Charged-current interaction of an energetic electron neutrino
 in a bubble chamber \protect\cite{wachsmuth}}
    \label{F:cgru:14f}
  \end{center}
\end{figure}

Even higher resolutions are obtained in cryogenic detectors. Figure~\ref{F:cgru:14e} 
\cite{CRESST}
shows the energy resolution for characteristic $X$ rays of Al, Ti, and Mn in a
superconducting phase-transition thermometer. The change of resistance upon an
energy absorption in the thermometer is transformed into a magnetic-flux 
change which is read by a superconducting quantum interference device (SQUID).
 
 
\section*{Interaction of Neutrons}
\noindent In the same way as photons are detected via their interactions also
neutrons have to be measured indirectly. Depending on the neutron energy
various reactions can be considered which produce charged particles
which are then detected via their ionization or scintillation 
\cite{r:cgru:[2]}.
\begin{itemize}
\item[a)]
Low energies ($< 20\,$MeV)
\beqar
n + {^6{\rm Li}} & \rightarrow & \alpha + {^3{\rm H}} \nonumber \\
n + {^{10}{\rm B}} & \rightarrow & \alpha + {^7{\rm Li}} \nonumber \\
n + {^3{\rm He}} & \rightarrow & p + {^3{\rm H}}  \\
n + p & \rightarrow & n+p \nonumber
\eeqar
The conversion material can be a component of a scintillator (e.g.\ LiI (Tl)),
a
thin layer of material in front of the sensitive volume of a gaseous detector
 (boron layer), or an admixture to the counting gas of a proportional counter
(BF$_3$,  $^3$He, or protons in CH$_4$).
\item[b)]
Medium energies ($20\,{\rm MeV} \leq E_{\rm kin} \leq 1\,$GeV) \\
The ($n,p$)-recoil reaction can be used for neutron detection in detectors
which contain many quasi-free protons in their sensitive volume (e.g.\
hydrocarbons).
\item[c)]
High energies ($E > 1\,$GeV) \\
Neutrons of high-energy initiate hadron cascades in inelastic interactions
which are easy to identify in hadron calorimeters.
\end{itemize}

Neutrons are detected with relatively high efficiency at very low energies.
Therefore, it is often useful to slow down neutrons with substances
containing many protons, because neutrons can transfer a large amount of
energy to collision partners of the same mass. 
%old
%Figure \ref{F:cgru:39} shows the cross
%sections for neutron-induced reactions in the energy range from 1\,eV to
%1\,MeV \cite{r:cgru:[2]}. 
%end old
In some fields of application, like in radiation 
protection at
nuclear reactors, it is of importance to know the energy of fission
neutrons, because  the relative  biological effectiveness depends on it.
This can e.g.\ be achieved with a stack of plastic  detectors interleaved with
foils of materials with different threshold energies for neutron conversion
\cite{r:cgru:[28]}.

\section*{Interactions of Neutrinos}
\noindent Neutrinos are very difficult to detect. Depending on the neutrino flavor the
following inverse beta decay like interactions can be considered:
\beqar
\nu_e + n & \rightarrow & p + e^- \nonumber \\
\bar{\nu}_e + p & \rightarrow & n + e^+ \vspace{1mm}\nonumber \\
\nu_{\mu} + n & \rightarrow & p + \mu^- \nonumber \\
\bar{\nu}_{\mu} + p & \rightarrow & n + \mu^+ \vspace{1mm}\\
\nu_{\tau} + n & \rightarrow & p + \tau^- \nonumber \\ 
\bar{\nu}_{\tau} + p & \rightarrow & n + \tau^+ \nonumber
\eeqar
The cross section for $\nu_e$ detection in the MeV range can be
estimated as \protect\cite{r:cgru:[32]}
\beqar
\sigma (\nu_eN) & = & \frac{4}{\pi} \cdot 10^{-10} \left( \frac{\hbar p}{(m_p
c)^2} \right)^2  \nonumber \\
& = & 6.4 \cdot 10^{-44}\,{\rm cm}^2 \mbox{ for 1\,MeV} \quad.
\eeqar
This means that the interaction probability of e.g.\ solar neutrinos in a water
Cherenkov counter of $d=100$\,meter  thickness is only
\beq
\phi = \sigma \cdot N_{\rm A} \cdot d = 3.8 \cdot 10^{-16} \quad.
\eeq
Since the coupling constant of weak interactions has a dimension of
1/GeV$^2$, the neutrino cross section must rise at high energies like the
square of the center--of--mass energy. For fixed-target experiments we can
parametrize
\beqar
\sigma (\nu_{\mu} N) & = & 0.67 \cdot 10^{-38} E_{\nu} [{\rm GeV}] \quad {\rm
cm}^2/{\rm nucleon} \nonumber \\
\sigma (\bar{\nu}_{\mu} N) & = & 0.34 \cdot 10^{-38} E_{\nu} [{\rm GeV}] \quad
{\rm cm}^2/{\rm nucleon}
\eeqar
This shows that even at 100\,GeV the neutrino cross section is lower by 11
orders of magnitude compared to the total proton--proton cross section.

Figure~\ref{F:cgru:14f} \cite{wachsmuth} shows the interaction of a high-energy electron neutrino
in a bubble chamber ($\nu_{e} + nucleon \to e^{-} + hadrons $) producing an 
electron in a charged-current interaction. The electron initiates an 
electromagnetic cascade in the bubble chamber.


\section*{Electromagnetic Cascades}
\noindent The development of cascades induced by electrons, positrons, or photons
is governed by bremsstrahlung of electrons and pair production of
photons. Secondary particle production continues until photons fall below
the pair production threshold, and energy losses of electrons other than
bremsstrahlung start to dominate: the number of shower particles decays
exponentially.

Already a very simple model can describe the main features
of particle multiplication in electromagnetic cascades: a photon
of energy $E_0$ starts the cascade by producing an $e^+e^-$ pair after
one radiation length. Assuming that the energy is shared symmetrically
between the particles at each multiplication step, one gets at the depth $t$
\beq
N(t) = 2^t
\eeq
particles with energy
\beq
E(t) = E_0 \cdot 2^{-t} \quad.
\eeq
The multiplication continues until the electrons fall below the critical
energy $ E_{\rm c}$
\beq
 E_{\rm c} = E_0 \cdot 2^{-t_{\rm max}} \quad.
\label{E:cgru:112}
\eeq
From then on ($t > t_{\rm max}$) the shower particles are only absorbed. The
position of the shower maximum is obtained from Eq.~(\ref{E:cgru:112})
\beq
t_{\rm max} = \frac{\ln E_0/ E_{\rm c}}{\ln 2} \propto \ln E_0 \quad.
\eeq
The total number of shower particles is
\beqar
S & = & \sum^{t_{\rm max}}_{t=0} N(t) = \sum 2^t = 2^{t_{\rm max}+1} - 1 \approx
2^{t_{\rm max}+1} \nonumber \\
S & = & 2 \cdot 2^{t_{\rm max}} = 2 \cdot \frac{E_0}{ E_{\rm c}} \propto E_0 \quad.
\eeqar
If the shower particles are sampled in steps $t$ measured in  units of
$X_0$, the total track length is obtained as
\beq
S^{\ast} = \frac{S}{t} = 2 \frac{E_0}{ E_{\rm c}} \cdot \frac{1}{t} \quad,
\eeq
which leads to an energy resolution of
\beq
\frac{\sigma}{E_0} = \frac{\sqrt{S^{\ast}}}{S^{\ast}} = \frac{\sqrt{t}}{\sqrt{2
E_0/ E_{\rm c}}} \propto \frac{\sqrt{t}}{\sqrt{E_0}} \quad.
\eeq
In a more realistic description the longitudinal development of the electron
shower can be approximated by \cite{r:cgru:[6]}
\beq
\frac{{\rm d}E}{{\rm d}t} = \mbox{const} \cdot t^a  \cdot {\rm e}^{-bt} \quad,
\eeq
where $a$, $b$ are fit parameters.

%new
Figures~\ref{F:cgru:new14} \cite{r:cgru:[new34]} and \ref{F:15a} \cite{contemp}  show muon-induced 
electromagnetic cascades in a multi-plate cloud chamber.

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.7\textwidth
%,angle=-90.0
     ,clip]
%{bild42}
{fig15compr}
\vspace*{0.5cm}
    \caption{Some muon-induced electromagnetic cascades in a
      multi-plate cloud chamber operated in a concrete-shielded air-shower 
      array \protect\cite{r:cgru:[new34]}}
    \label{F:cgru:new14}
  \end{center}
\end{figure}
%end new
\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.6\textwidth,clip]
%ba 20_7 {fig15a}
{h38compr}
    \caption{Muon induced electromagnetic cascade in a multiplate cloud chamber \protect\cite{contemp}}
    \label{F:15a}
  \end{center}
\end{figure}
The lateral spread of an electromagnetic shower is mainly caused by
multiple scattering. It is described by the Moli\`{e}re radius
\beq
R_{\rm m} = \frac{21\,{\rm MeV}}{ E_{\rm c}} X_0 \: [{\rm g/cm}^2] \quad.
\eeq
95\% of the shower energy in a homogeneous calorimeter is contained in a
cylinder of radius $ 2 R_{\rm m}$ around the shower axis.

Figure~\ref{F:cgru:45} demonstrates the interplay of the longitudinal
and lateral
development of an electromagnetic shower \cite{r:cgru:[2]}.

%\clearpage
\begin{figure}[!ht]
\begin{center}
    \includegraphics[width=0.85\textwidth,angle=0.0
     ,clip]
%{bild45}
{h39compr}
    \vspace*{0.2cm}
    \caption{Sketch of the longitudinal  and lateral development of an
    electromagnetic cascade in a homogeneous absorber 
    \protect\cite{r:cgru:[2]}}
    \label{F:cgru:45}
\end{center}
\end{figure}

\newpage
\section*{Hadron Cascades}
\noindent The longitudinal development of electromagnetic cascades is
characterized by the radiation length $X_0$ and their lateral width is
determined by multiple scattering. In contrast to this, hadron showers are
governed in their longitudinal structure by the nuclear interaction length
$\lambda$ and by transverse momenta of secondary particles as far as
lateral width is
concerned. Since for most materials $\lambda \gg X_0$, and $\langle
p_{\rm T}^{\rm interaction}\rangle \gg \langle p_{\rm T}^{\rm multiple\:
scattering}\rangle$ hadron showers are longer and wider.

Part of the energy of the incident hadron is spent to break up nuclear
bonds. This fraction of the energy is invisible in hadron calorimeters.
Further energy is lost by escaping particles like neutrinos and muons as a
result of hadron decays. Since the fraction of lost binding energy and
escaping particles fluctuates considerably, the energy resolution of hadron
calorimeters is systematically inferior to electron calorimeters.

%new
Figure~\ref{F:gruphand} sketches the various fractions of energy in a hadronic cascade and
their variation with the hadron energy.

The longitudinal development of pion-induced hadron cascades is plotted in
Fig.~\ref{F:cgru:new16} \cite{r:cgru:[37]}.
Figure \ref{F:cgru:new17} 
shows a comparison between proton-, iron-, and
photon-induced cascades in the atmosphere \cite{r:cgru:knapp}.

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[angle=0.5
%,totalheight=8cm
,width=.6\textwidth]
%          {bild47}
{h26}
    \vspace{5pt}
    \caption{Fractions of the total energy in a hadronic cascade that go into
     nuclear fragments, binding energy, charged particles, and electromagnetic
      cascades in their variation with energy}
    \label{F:gruphand}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[angle=0.5
%,totalheight=8cm
,width=.5\textwidth]
%          {bild47}
{hcascadecompr}
    \vspace{5pt}
    \caption{Longitudinal development of pion-induced hadron cascades 
      \protect\cite{r:cgru:[37]}}
    \label{F:cgru:new16}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[width=0.7\textwidth,clip]
%{bild48}
{fig17newcompr}
\vspace*{0.2cm}
    \caption{Comparison between proton-, iron-, and photon-induced
      cascades in the atmosphere. The primary energy in each case is
      10$^{14}$eV \protect\cite{r:cgru:knapp}.}
    \label{F:cgru:new17}
  \end{center}
\end{figure}
%end new

The different response of calorimeters to electrons and hadrons is an
undesirable feature for the energy measurement of jets of unknown
particle composition. By appropriate compensation techniques, however,
the electron--to--hadron response can be equalized.

\section*{Particle Identification}
\noindent Particle identification is based on measurements which are sensitive
to the particle velocity, its charge, and its momentum.
Figure \ref{F:cgru:new18} sketches the different possibilities to
separate photons, electrons, positrons, muons, charged pions, protons, 
neutrons, and neutrinos in a mixed particle beam using a general-purpose detector.

\begin{figure}[ht!]
  \begin{center}
     \includegraphics[bb=50 35 760 515,width=.7\textwidth,clip]{fig18compr}
    \caption{Particle identification using a detector consisting of a
      tracking chamber, Cherenkov counters, calorimetry, and muon chambers.}
    \label{F:cgru:new18}
  \end{center}
\end{figure}

One particle identification technique that has not been discussed so far in 
detail is the time--of--flight measurement (TOF). Two particles of different velocity
$v_{1}$ and $v_{2}$ passing through a telescope consisting of two scintillation counters at a distance $L$ will exhibit a time--of--flight difference
\beq
\Delta t = L \cdot \bigg( \frac{1}{v_{1}} - \frac{1}{v_{2}} \bigg)
= \frac{L}{c} \cdot \bigg(\frac{1}{\beta_{1}} - \frac{1}{\beta_{2}} \bigg) \quad .
\label{102}
\eeq
Replacing the normalized velocities $\beta_{i} = v_{i}/c$ by the corresponding
Lorentz factors $\gamma_{i} = 1/\sqrt{1-\beta_{i}^{2}}$ in Eq.~(\ref{102}) one 
gets 
\beq
\Delta t = \frac{L}{c} \cdot \Bigg\{ \sqrt{\frac{\gamma_{1}^{2}}{\gamma_{1}^{2} -1}} - \sqrt{\frac{\gamma_{2}^{2}}{\gamma_{2}^{2} -1}} 
\Bigg\} \quad .
\label{103}
\eeq
For relativistic particles ($E \gg m_{0} \cdot c^{2}$), Eq.~(\ref{103}) can be 
approximated by
\beq
\Delta t = \frac{L}{c} \cdot \Bigg\{ \sqrt{1 + \bigg( \frac{m_{1} \cdot c^{2}}{E_{1}}\bigg)^{2}} - \sqrt{1 + \bigg( \frac{m_{2} \cdot c^{2}}{E_{2}}\bigg)^{2}} 
\Bigg\} \quad .
\eeq
Since in the relativistic case $E \approx p \cdot c $, one gets for a momentum-defined
 beam ($E_{1}\approx p_{1} \cdot c$, $ E_{2}\approx p_{2} \cdot c$, 
 $p_{1} = p_{2} = p $)
\beq
\Delta t = \frac{L \cdot c}{2 \cdot p^{2}} \cdot \big(m_{1}^{2}-m_{2}^{2}\big)
\eeq 
allowing the identification of particles with different mass.
Figure~\ref{F:cgru:19a} \cite{fragiacomo} shows the particle identification power in a scintillator 
system using TOF and d$E$/d$x$ information in a momentum-defined beam containing
electrons, muons and pions of 107.5\,MeV/$c$.
%\begin{figure}[ht!]
%  \begin{center}
%\vspace*{-0.5cm}
%\includegraphics[width=1.\textwidth,clip]
%    {fig19acomprnew}
%    \caption{e/$\mu$/$\pi$ separation in a momentum-defined beam 
% ($p = 107.5$\,MeV/$c$) using TOF and d$E$/d$x$ techniques \protect\cite{fragiacomo}.}
%    \label{F:cgru:new19}
%  \end{center}
%\end{figure}

\begin{figure}[ht!]
  \begin{center}
\vspace*{0.3cm}
%\subfigure{\epsfig{file=fig19atofcompr,width=0.45\textwidth}}
\includegraphics[height=6.5cm,width=0.475\textwidth,clip]
    {fig19atofcompr}
\hspace*{0.3cm}
%\subfigure{\epsfig{file=fig19adedxcompr,width=0.45\textwidth}}
\includegraphics[height=6.5cm,width=0.475\textwidth,clip]
    {fig19adedxcompr}
%\caption{Particle identification in a balloon-borne experiment using
%      momentum, time--of--flight, d$E$/d$x$, and Cherenkov radiation
%      information \protect\cite{r:cgru:Mitchell}.}
    \caption{e/$\mu$/$\pi$ separation in a momentum-defined beam 
 ($p = 107.5$\,MeV/$c$) using TOF and d$E$/d$x$ techniques \protect\cite{fragiacomo}.}
    \label{F:cgru:19a}
  \end{center}
\end{figure}
\begin{figure}[ht!]
  \begin{center}
\vspace*{-0.5cm}
\includegraphics[height=6.cm,width=0.7\textwidth,clip]
    {B4compr}
 \caption{Pion/proton separation in a 2\,GeV/c beam with a scintillator TOF 
system. The distance of the scintillation counters was 1.3\,m \protect\cite{BELLE}.}
    \label{F:cgru:19b}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
\vspace*{-0.5cm}
\includegraphics[width=0.7\textwidth,clip]
    {B6compr}
\caption{Time resolution in a multi-gap resistive plate chamber \protect\cite{Sauli,ALICE}.}
    \label{F:cgru:19c}
  \end{center}
\end{figure}
Essential for an effective particle identification with TOF techniques is an
excellent time resolution. Figures~\ref{F:cgru:19b} and \ref{F:cgru:19c} show the time resolution in a 
scintillator TOF system and a multi-gap resistive plate chamber \cite{BELLE,Sauli,ALICE} 
allowing for an efficient $e/\pi/p$ separation. 
Figures~\ref{F:cgru:20a} \cite{r:cgru:Kremer} and  \ref{F:cgru:new19} \cite{r:cgru:Mitchell} show the particle separation power of a
balloon-borne experiment using momentum, time--of--flight, d$E$/d$x$ and
Cherenkov-radiation measurements for singly charged 
particles at an altitude of 1234\,m (ground level) and at flight altitudes
 ($\approx 40$\,km).

\begin{figure}[ht!]
  \begin{center}
\vspace*{-0.5cm}
\includegraphics[width=0.55\textwidth,clip]
%    {folie72left}
{rigiditycompr}
    \caption{Particle composition in a balloon-borne experiment at ground level
     (altitude 1234 m) \protect\cite{r:cgru:Kremer}. Rigidity is defined as momentum
     divided by the charge of the particle. }
    \label{F:cgru:20a}
  \end{center}
\end{figure}
\begin{figure}[ht!]
  \begin{center}
\vspace*{-0.5cm}
\includegraphics[width=0.55\textwidth,clip]
    {bild55compr}
    \caption{Particle identification in a balloon-borne experiment using
      momentum, time--of--flight, d$E$/d$x$ and Cherenkov radiation
      information \protect\cite{r:cgru:Mitchell}.}
    \label{F:cgru:new19}
  \end{center}
\end{figure} 

\begin{figure}[ht!]
  \begin{center}
\vspace*{-0.5cm}
\includegraphics[width=0.55\textwidth,clip]
    {folie73leftcompr}
    \caption{Identification of proton and helium nuclei in primary cosmic rays 
    in a balloon-borne experiment \protect\cite{Reimer}}
    \label{F:Reimer}
  \end{center}
\end{figure}


Relatively easy is the proton/helium separation in such an experiment (Fig.~\ref
{F:Reimer} \cite{Reimer}).

Even the abundance of different helium isotopes can be determined from 
a velocity and momentum measurement (Fig.~\ref{F:cgru:new20}
\cite{r:cgru:Simon}).
This is feasible, because at fixed momentum the lighter isotope $^3$He 
is faster than the more abundant $^4$He.

\begin{figure}[ht!]
  \begin{center}
    \includegraphics[
bb=30 148 410 402,
width=0.7\textwidth,
%totalheight=7cm
,clip]
%{br_he3_new}
{B_R_HE3_2_new}
    \caption{Isotopic abundance of energetic cosmic-ray helium nuclei
      \protect\cite{r:cgru:Simon}}
    \label{F:cgru:new20}
  \end{center}
\end{figure}


\begin{figure}[ht!]
  \begin{center}
\vspace*{-0.5cm}
\subfigure {\psfig{file=goebel1,width=0.65\textwidth}}
\subfigure {\psfig{file=goebel2,width=0.65\textwidth}}
%\includegraphics[width=0.65\textwidth,clip]
%    {goebel}
    \caption{Identification of light elements in primary cosmic rays in a 
balloon experiment \protect\cite{goebel}.}
    \label{F:goebelisomax98}
  \end{center}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
\vspace*{-0.5cm}
\includegraphics[width=0.6\textwidth,clip]
    {chargecompr}
    \caption{Chemical composition of primary cosmic rays as determined
     by range and d$E$/d$x$ measurements \protect\cite{Hesse}.}
    \label{F:charge}
  \end{center}
\end{figure}

 In very much the same way works the identification of the light elements
in primary cosmic rays (Fig.~\ref{F:goebelisomax98} \cite{goebel}) by using d$E$/d$x$ and TOF techniques.
 The elements from lithium up to oxygen can be resolved with high resolution.
A similar method can be used to extend the charge spectrum up to the iron family
 (Fig.~\ref{F:charge} \cite{Hesse}) 

\clearpage
\section*{Conclusion}
\noindent Basic physical principles can be used to identify all kinds of  elementary
particles and nuclei. The precise measurement of the particle composition
in high-energy physics experiments at accelerators and in cosmic rays is
essential for the insight into the underlying physics processes. If a particle
 cannot be directly identified -- such as is the case for short-lived particles like 
$K_{s}^{0}$ or $\Lambda^{0}$ -- their decay products can be measured and 
identified. Knowing their energy, momentum, and mass, the four-vectors of the
 decay products can be combined to form the invariant mass of the parent 
particle ($K_{s}^{0} \to \pi^{+} + \pi^{-} $, $\Lambda^{0} \to p + \pi^{-}$).
 This technique has become very powerful especially for the tagging of very 
 short-lived mesons like $B^{0}$ or $D^{0}$ in hadronic jets.

Neutrinos or other weakly interacting neutral particles are almost impossible
to identify directly. If, however, an event is fully contained in a detector
 and the center--of--mass energy is known the direction of the missing momentum
can be associated with the direction of the weakly interacting neutral 
particle, thereby identifying it. This technique has been used intensively
e.g. for the reconstruction of the leptonic decays of the charged vector 
bosons $W^{+}$ and $W^{-}$( $W^{+} \to e^{+} + \nu_{e}$, 
$W^{+} \to \mu^{+} + \nu_{\mu}$).
\section*{Acknowledgements}
\noindent It is a pleasure to thank the organizers of the Instrumentation School 
for their kind invitation and special hospitality. I would also like to thank
Dipl.\ Phys.\ Tilo Stroh and in particular Dipl.\ Phys.\ Anbang Ngac for their 
help in preparing the manuscript.
 
\begin{references}

\bibitem{r:cgru:[1]} K.\ Kleinknecht, {\it Detectors for Particle Radiation}, 
  Cambridge University Press 1998

\bibitem{r:cgru:[2]} C.\ Grupen, {\it Particle Detectors}, Cambridge 
  University Press 1996

\bibitem{r:cgru:[3]} R.\ Fernow, {\it  Introduction to Experimental Particle 
    Physics},
  Cambridge University Press 1989

\bibitem{Green} D.\ Green, {\it  The Physics of Particle Detectors},
  Cambridge University Press 2001 

\bibitem{r:cgru:[4]} W.R.\ Leo, {\it Techniques for Nuclear and Particle 
    Physics Experiments}, Springer, Berlin 1987

\bibitem{r:cgru:[5]}  B.\ Rossi, {\it High Energy Particles}, Prentice-Hall 
  (1952)

\bibitem{r:cgru:[6]} Particle Data Group, R.M.\ Barnett et al., {\it Phys.\ Rev.}
  {\bf D54} (1996) 1; {\it Eur.\ Phys.\ J.} {\bf C3} (1998) 1; 
  {\it Eur.\ Phys.\ J.} {\bf C15} (2000) 1 

\bibitem{Ruth} E.\ Rutherford, {\it Phil.\ Mag.\ 21} (1911) 669

\bibitem{Geiger} H.\ Geiger, E.\ Marsden , {\it Phil.\ Mag.\ 25} (1913) 604 

\bibitem{Hanson} A.O.\ Hanson et al., {\it Phys.\ Rev.} {\bf 84} (1951) 634 
 
\bibitem{Birkhoff} R.O.\ Birkhoff in {\it Handbook of Physics} XXXIV (1958) 

\bibitem{r:cgru:[7]} W.W.M.\ Allison, P.R.S.\ Wright, {\it Oxford Univ.\ 
Preprint} {\bf 35/83}
(1983) and \\
W.W.M.\ Allison, J.H.\ Cobb, {\it Ann.\ Rev.\ Nucl.\ Sci.} {\bf 30} (1980) 253

\bibitem{r:cgru:[newNM]} D.R.\ Nygren, J.N.\ Marx, {\it Physics Today} {\bf 31} (1978) 46
 
\bibitem{r:cgru:[8]} C.\ Grupen, {\it Ph.D.\ Thesis}, University of Kiel 1970

    
\bibitem{Titt} U.\ Titt et al., {\it Nucl.\ Instr.\ Meth.} {\bf A 416} (1998) 85 

\bibitem{Fraga} F.A.F\ Fraga et al., {\it Nucl.\ Sci.\ Symp. and Medical Imaging Conf.}, Seattle, Washington, Oct. 1999, to be publ. in {\it IEEE Trans.\ Nucl.\ Sci.}

\bibitem{Rochester} G.D\ Rochester, {\it Advancement of Science}, Dec.\ 1970, 183--194

\bibitem{Kraft} G.\ Kraft, {\it GSI-Preprint}\ {\bf 96--60} (1996) and 
private communication; {\it Nucl.\ Phys.} {\bf A 654} (1999) 1058c

\bibitem{bang} A.\ Ngac, {\it Diploma Thesis}, University of Siegen 2000

\bibitem{r:cgru:[10]} D.H.\ Wilkinson, {\it Nucl.\ Instr.\ Meth.}\ {\bf A 383} (1996) 513
\bibitem{r:cgru:[11]} L.D.\ Landau, {\it J.\ Exp.\ Phys.\ (USSR)} {\bf 8} (1944) 201
\bibitem{r:cgru:[12]} S.\ Behrens, A.C.\ Melissinos, {\it Univ.\ of 
Rochester Preprint} {\bf UR-776} (1981)

\bibitem{Moeller} S.P.\ M\"oller, {\it CERN} {\bf 94--05} (1994)

\bibitem{Gemmel} D.S.\ Gemmel, {\it Rev.\ Mod.\ Phys.}\ {\bf 46} (1974) 129 

\bibitem{r:cgru:[13]} K.\ Affholderbach et al., {\it Nucl.\ Instr.\ Meth.}
  {\bf A 410} (1998) 166
\bibitem{r:cgru:[17]} R.C.\ Fernow, {\it Brookhaven Nat.\ Lab.\  Preprint} 
{\bf BNL-42114} 
(1988)
\bibitem{r:cgru:[18]} J.\ Birks, {\it Theory and Practice of Scintillation 
Counting},
MacMillan 1964
\bibitem{BELLE} A.\ Satpathy et al., BELLE Collaboration, hep-ex/9903045 (1999)
\bibitem{r:cgru:Adam} I.\ Adam et al.\  SLAC-Pub-7706, Nov.\ 1997; and
  hep-ex/9712001 

\bibitem{r:cgru:Debbe} R.\ Debbe et al.\ hep-ex/9503006 (1995)

\bibitem{skamio} Super-Kamiokande home page \\ \verb@http://www.ps.uci.edu/~tomba/sk/tscan/pictures.html@

\bibitem{sno} Sudbury Neutrino Observatory home page \\ 
{\verb@http://www.sno.phy.queensu.ca/event/events.html@}

\bibitem{r:cgru:[21]} S.\ Paul, {\it CERN-PPE} {\bf 91--199} (1991)

\bibitem{normad} NOMAD collaboration, \\{\verb@http://nomadinfo.cern.ch/Public/detect/trd/trd.html@}\\
                 G.\ Bassompierre et al. {\it Nucl.\ Instr.\ Meth.} {\bf A 403} (1998) 363--382

\bibitem{aleph} ALEPH Collaboration (1998), unpublished

\bibitem{r:cgru:Close} F.\ Close, M.\ Marten, C.\ Sutton, {\it The Particle
  Explosion}, Oxford University Press 1987

%\bibitem{r:cgru:[28]}  E.\ Sauter, {\it Grundlagen des Strahlenschutzes}, 
%Thiemig, M\"unchen 1982

\bibitem{r:cgru:[30]}  Harshaw Chemical Company (H.\ Lentz, L.\ van Gelderen, 
Th.\ Courbois) 1969

\bibitem{tennessee} The University of Tennessee, Department of Physics and 
Astronomy,\\ {\verb@http://electron5.phys.utk.edu/gamma/detector.htm@}

\bibitem{Amptek} Amptek Inc. Bedford MA, USA; {\verb@http://www.amptek.com/gammax.html@}

\bibitem{hausmann} J.\ Hausmann, University of Siegen, private communication (2001).

\bibitem{CRESST} CRESST Collaboration, {\it Nucl.\ Instr.\ Meth.} {\bf A 354} (1995) 408

\bibitem{r:cgru:[28]}  E.\ Sauter, {\it Grundlagen des Strahlenschutzes}, 
Thiemig, M\"unchen 1982

\bibitem{r:cgru:[32]} D.H.\ Perkins {\it Introduction to High Energy Physics},
Addison-Wesley, 1986

\bibitem{wachsmuth} H.\ Wachsmuth, CERN, private communication (1998)

%\bibitem{r:cgru:[37]} M.\ Holder et al., {\it Nucl.\ Instr.\ Meth.} {\bf 151} 
%69 (1978)

%\bibitem{r:cgru:Adam} I.\ Adam et al.\  SLAC-Pub-7706, Nov.\ 1997; and
%  hep-ex/9712001 

%\bibitem{r:cgru:Debbe} R.\ Debbe et al.\ hep-ex/9503006 (1995) 

\bibitem{r:cgru:[new34]} W.\ Wolter, private communication 1999

\bibitem{contemp} G.D.\ Rochester, K.E.\ Turvev, {\it Contemp.\ Phys.} {\bf 22} 
(1981) 425

\bibitem{r:cgru:knapp} J.\ Knapp, D.\ Heck
  {\it Luftschauer-Simulationsrechnungen mit dem Corsika-Programm}
  Forschungszentrum Karlsruhe Nachrichten {\bf 30} (1998) 27

\bibitem{r:cgru:[37]} M.\ Holder et al., {\it Nucl.\ Instr.\ Meth.} {\bf 151} 
69 (1978)
 
\bibitem{fragiacomo} E.\ Fragiacomo et al., {\it Nucl.\ Instr.\ Meth.} {\bf A 439} 
 (2000) 45
  
\bibitem{Sauli} F.\ Sauli, {\it CERN-EP/2000-080} 

\bibitem{ALICE} ALICE technical design report, {\it CERN/LHCC 2000-12} (2000)

\bibitem{r:cgru:Kremer} J.\ Kremer, {\it Ph.D.\ Thesis}, University of Siegen 1999

\bibitem{r:cgru:Mitchell} J.\ W.\ Mitchell et al., {\it Phys.\ Rev.\ Lett.}
  {\bf 76} (1996) 3057

\bibitem{Reimer} O.\ Reimer, {\it Ph.D.\ Thesis}, University of Siegen 1995\\
  J.\ W.\ Mitchell et al., {\it Proc.\ Cosmic Ray Conf.\ \rm Rome} 1995 

\bibitem{r:cgru:Simon} M.\ Simon, H.\ G\"obel, private communication 1999

\bibitem{goebel} H.\ G\"obel, {\it Ph.D.\ Thesis}, University of Siegen 2001 

%\bibitem{r:cgru:Close} F.\ Close, M.\ Marten, C.\ Sutton, "The Particle
%  Explosion", Oxford University Press 1987

\bibitem{Hesse} A.\ Hesse, {\it Proc.\ 22nd Int.\ Cosmic Ray Conf.\ \rm Dublin} 
{\bf Volume 1} (1991) 596

\end{references}

\end{document}






