\begin{figure}
	\centering
	\includegraphics[width=0.8\textwidth]{Images/pdf/rich}
	\caption{Draft of a simple RICH detector. The radius of the circle in the focal plane is determined by the velocity of the particle, while the position of its centre depends on the particle direction.}\label{img:rich}
\end{figure}

\figurename~\ref{img:rich} describes how a RICH detector works.
%``\textbf{RICH}'' stands for \textit{Ring Imaging \v{C}erenkov} detector.\\
%
When a particle goes through a medium at a velocity $\beta = v/c > 1/n$, where $n$ is the refractive index of the medium, it emits \v{C}erenkov light at an angle $\theta_c$ relative to the particle trajectory, such that
%``RICH'' stands for \textit{Ring Imaging \v{C}erenkov} detector. This device is based on the \v{C}erenkov emission of light by a charged particle going through a medium at a speed $\beta > 1/n$, where $n$ is the refractive index of the medium. Light is emitted at an angle $\theta_C$ relative to the particle direction, defined by
\begin{align}
\cos{\theta_c} = \dfrac{1}{n \, \beta}\label{eqn:beta-thetac}
\end{align}
thus forming what is called a \textit{\v{C}erenkov cone}. It follows that a threshold $\beta_{th}$ exists below which no radiation is emitted:
\begin{align}
&\beta_{th} = \frac{1}{n} \qquad \left( \theta_c = 0 \right) \label{eqn:cher_betath}
\end{align}
while the maximum angle of emission is achieved for $\beta \rightarrow 1$:
\begin{align}
&\cos\theta_{max} \rightarrow \frac{1}{n} \qquad \left( \beta \rightarrow 1 \right)
\end{align}

From Eqn.~\ref{eqn:cher_betath} we derive the threshold momentum $P_{th}$ for a particle of mass $m$ to emit \v{C}erenkov radiation:
\begin{align}
&P_{th}(m) = \frac{m}{\sqrt{n^2 -1}} \label{eqn:cher_pth}
\end{align}

The light cone is projected on a focal plane, perpendicular to the beam direction, by means of a spherical mirror (or a system of mirrors, as it will be described in the following section) of focal length $f$. For particles travelling parallel to the beam line, the resulting image on the focal plane is a ring of radius
\begin{align}
r_c = f \, \tan {\theta_c}\label{eqn:rc-thetac}
\end{align}
while, for particles travelling at an angle $\theta$ to the beam line, the same image appears shifted by a distance
\begin{align}
d = f \, \tan {\theta}\label{eqn:d-theta}
\end{align}
from the focus. \figurename~\ref{img:rich} shows a sketch of a basic RICH detector.\\

Larger rings on the focal plane correspond to particles crossing the RICH radiator volume at a larger velocity (keeping the type of the particle fixed). On the other hand, if the momentum of the beam is known to an adequate precision, the radius of the reflected \v{C}erenkov ring can be used to compute the mass of the crossing particle, and therefore to perform P.ID. (\textit{Particle Identification}).\\

The following relation holds:
\begin{align}
P(\theta_c) &= \frac{m}{n} \, \frac{1}{\sqrt{\sin^2\theta_{max} - \sin^2\theta_c}}
\end{align}
and therefore:
\begin{align}
P(r_c) &= m \, \frac{\sqrt{f^2 + r_c^2}}{\sqrt{r^2_{max} - r^2_c}} \simeq \frac{m\,f}{\sqrt{r^2_{max} - r^2_c}}\label{eqn:rich-p}
\end{align}
where $ r_{max} = f \sqrt{n^2 -1}$.
%Given the momentum of a particle, %by means of a RICH detector it is possible 
%a RICH detector allows 
%to compute its mass, and thus to identify it. In fact, the aims of this device in the frame of the NA62 experiment are:
%\begin{itemize}
%\item separating muons from pions: in the momentum range from 15 GeV to 35 GeV the separation is achieved with a precision of $10^{-2}$;
%\item generating a level 0 trigger for charged particles;
%\end{itemize}
%
%DA FINIRE - DA RIFARE
%