Subfield: Optics

CHERENKOV LIGHT AND RING IMAGING COUNTER 

Light propagates in vacuum with the speed $c$. There is no particle which moves with a speed higher than $c$. However, it is possible that in a transparent medium a particle moves with a speed $v$ higher than the speed of the light in the same medium $\frac{c}{n}$, where $n$ is the refraction index of the medium. Experiment (Cherenkov, 1934) and theory (Tamm and Frank, 1937) showed that a charged particle, moving with a speed $v$ in a transparent medium with refractive index $n$ such that $v>\frac{C}{n}$, radiates light, called Cherenkov light, in directions forming with the trajectory an angle

$$
\theta=\arccos \frac{1}{\beta n}
\tag{1}
$$

where $\beta=\frac{v}{c}$.

<img_4441>

1. To establish this fact, consider a particle moving at constant velocity $v>\frac{C}{n}$ on a straight line. It passes $\mathrm{A}$ at time 0 and $\mathrm{B}$ at time $t_{1}$. As the problem is symmetric with respect to rotations around $\mathrm{AB}$, it is sufficient to consider light rays in a plane containing $\mathrm{AB}$.

At any point $\mathrm{C}$ between $\mathrm{A}$ and $\mathrm{B}$, the particle emits a spherical light wave, which propagates with velocity $\frac{c}{n}$. We define the wave front at a given time $t$ as the envelope of all these spheres at this time.
Context question:
1.1. Determine the wave front at time $t_{1}$ and draw its intersection with a plane containing the trajectory of the particle.
Context answer:
<img_4447>


Context question:
1.2. Express the angle $\varphi$ between this intersection and the trajectory of the particle in terms of $n$ and $\beta$.
Context answer:
\boxed{$\varphi=\arcsin \frac{1}{\beta n}$}


Context question:
2. Let us consider a beam of particles moving with velocity $v>\frac{C}{n}$, such that the angle $\theta$ is small, along a straight line IS. The beam crosses a concave spherical mirror of focal length $f$ and center $\mathrm{C}$, at point $\mathrm{S}$. SC makes with SI a small angle $\alpha$ (see the figure in the Answer Sheet). The particle beam creates a ring image in the focal plane of the mirror.


Explain why with the help of a sketch illustrating this fact. Give the position of the center $\mathrm{O}$ and the radius $r$ of the ring image.

This set up is used in ring imaging Cherenkov counters (RICH) and the medium which the particle traverses is called the radiator.

Note: in all questions of the present problem, terms of second order and higher in $\alpha$ and $\theta$ will be neglected.
Context answer:
<img_4436>


Extra Supplementary Reading Materials:

3. A beam of particles of known momentum $p=10.0 \mathrm{GeV} / \mathrm{c}$ consists of three types of particles: protons, kaons and pions, with rest mass $M_{\mathrm{p}}=0.94 \mathrm{GeV} / c^{2}$, $M_{\kappa}=0.50 \mathrm{GeV} / c^{2}$ and $M_{\pi}=0.14 \mathrm{GeV} / c^{2}$, respectively. Remember that $p c$ and $M c^{2}$ have the dimension of an energy, and $1 \mathrm{eV}$ is the energy acquired by an electron after being accelerated by a voltage $1 \mathrm{~V}$, and $1 \mathrm{GeV}=10^{9} \mathrm{eV}, 1 \mathrm{MeV}=10^{6} \mathrm{eV}$.

The particle beam traverses an air medium (the radiator) under the pressure $P$. The refraction index of air depends on the air pressure $P$ according to the relation $n=1+a P$ where $a=2.7 \times 10^{-4} \mathrm{~atm}^{-1}$
Context question:
3.1. Calculate for each of the three particle types the minimal value $P_{\min }$ of the air pressure such that they emit Cherenkov light.
Context answer:
$$
\begin{array}{ll}
P_{\min }=16 \mathrm{~atm} & \text { for protons, } \\
P_{\min }=4.6 \mathrm{~atm} & \text { for kaons, } \\
P_{\min }=0.36 \mathrm{~atm} & \text { for pions. }
\end{array}
$$


Context question:
3.2. Calculate the pressure $P_{\frac{1}{2}}$ such that the ring image of kaons has a radius equal to one half of that corresponding to pions. Calculate the values of $\theta_{\kappa}$ and $\theta_{\pi}$ in this case.

Is it possible to observe the ring image of protons under this pressure?
Context answer:
\boxed{$P_{\frac{1}{2}}=6 $ , $\theta_{\kappa}=1.6$ , $\theta_{\pi}=3.2$}


Extra Supplementary Reading Materials:

4. Assume now that the beam is not perfectly monochromatic: the particles momenta are distributed over an interval centered at $10 \mathrm{GeV} / \mathrm{c}$ having a half width at half height $\Delta p$. This makes the ring image broaden, correspondingly $\theta$ distribution has a half width at half height $\Delta \theta$. The pressure of the radiator is $P_{\frac{1}{2}}$ determined in 3.2.
Context question:
4.1. Calculate $\frac{\Delta \theta_{\kappa}}{\Delta p}$ and $\frac{\Delta \theta_{\pi}}{\Delta p}$, the values taken by $\frac{\Delta \theta}{\Delta p}$ in the pions and kaons cases.
Context answer:
\boxed{$\frac{\Delta \theta_{\kappa}}{\Delta p}=0.51 $ , $\frac{\Delta \theta_{\pi}}{\Delta p}=0.02$}


Context question:
4.2. When the separation between the two ring images, $\theta_{\pi}-\theta_{\kappa}$, is greater than 10 times the half-width sum $\Delta \theta=\Delta \theta_{\kappa}+\Delta \theta_{\pi}$, that is $\theta_{\pi}-\theta_{\kappa}>10 \Delta \theta$, it is possible to distinguish well the two ring images. Calculate the maximal value of $\Delta p$ such that the two ring images can still be well distinguished.
Context answer:
\boxed{$0.3 $}


Extra Supplementary Reading Materials:

5. Cherenkov first discovered the effect bearing his name when he was observing a bottle of water located near a radioactive source. He saw that the water in the bottle emitted light.
Context question:
5.1. Find out the minimal kinetic energy $T_{\min }$ of a particle with a rest mass $M$ moving in water, such that it emits Cherenkov light. The index of refraction of water is $n=1.33$.
Context answer:
$0.517 M c^{2}$


Context question:
5.2. The radioactive source used by Cherenkov emits either $\alpha$ particles (i.e. helium nuclei) having a rest mass $M_{\alpha}=3.8 \mathrm{GeV} / c^{2}$ or $\beta$ particles (i.e. electrons) having a rest mass $M_{\mathrm{e}}=0.51 \mathrm{MeV} / c^{2}$. Calculate the numerical values of $T_{\min }$ for $\alpha$ particles and $\beta$ particles.

Knowing that the kinetic energy of particles emitted by radioactive sources never exceeds a few $\mathrm{MeV}$, find out which particles give rise to the radiation observed by Cherenkov.
Context answer:
\boxed{1.96, 0.264}


Extra Supplementary Reading Materials:

6. In the previous sections of the problem, the dependence of the Cherenkov effect on wavelength $\lambda$ has been ignored. We now take into account the fact that the Cherenkov radiation of a particle has a broad continuous spectrum including the visible range (wavelengths from $0.4 \mu \mathrm{m}$ to $0.8 \mu \mathrm{m}$ ). We know also that the index of refraction $n$ of the radiator decreases linearly by $2 \%$ of $n-1$ when $\lambda$ increases over this range.

Question: 6.1. Consider a beam of pions with definite momentum of $10.0 \mathrm{GeV} / \mathrm{c}$ moving in air at pressure $6 \mathrm{~atm}$. Find out the angular difference $\delta \theta$ associated with the two ends of the visible range.

Solution:
For a beam of particles having a definite momentum the dependence of the angle $\theta$ on the refraction index $n$ of the medium is given by the expression

$$
\cos \theta=\frac{1}{n \beta}
\tag{14}
$$

Let $\delta \theta$ be the difference of $\theta$ between two rings corresponding to two wavelengths limiting the visible range, i.e. to wavelengths of $0.4 \mu \mathrm{m}$ (violet) and $0.8 \mu \mathrm{m}$ (red), respectively. The difference in the refraction indexes at these wavelengths is $n_{\mathrm{v}}-n_{\mathrm{r}}=\delta n=0.02(n-1)$.

Logarithmically differentiating both sides of equation (14) gives



$$
\frac{\sin \theta \times \delta \theta}{\cos \theta}=\frac{\delta n}{n}
\tag{15}
$$

Corresponding to the pressure of the radiator $P=6 \mathrm{~atm}$ we have from 4.2. the values $\theta_{\pi}=3.2^{\circ}, \quad n=1.00162$.

Putting approximately $\tan \theta=\theta$ and $n=1$, we get $\delta \theta=\frac{\delta n}{\theta}=0.033^{\circ}$.

Final answer: $0.033$ $^{\circ}$