4.1. Calculate and , the values taken by in the pions and kaons cases. · OlympiadBench #1332

Physics · optics

Problem

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. Given constants: 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}

. <image_1> 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: <image_2> 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:

\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: <image_3> 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:

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.

Related theory

optics

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