Astroparticle Physics

62 4 Physics of Particle and Radiation Detection3. The simplified energy loss of a muon is parameterizedby (4.8). Work out the range of a muon of energyE (= 100 GeV) in rock (ϱ rock = 2.5g/cm 3 ) underthe assumption that a (= 2MeV/(g/cm 2 )) and b(= 4.4 × 10 −6 cm 2 /g for rock) are energy independent.4. Show that the mass of a charged particle can be inferredfrom the Cherenkov angle θ C and momentum p bym 0 = p c√n 2 cos 2 θ C − 1 ,where n is the index of refraction.5. In a cryogenic argon calorimeter (T = 1.1 Kelvin, mass1 g) a WIMP (weakly interacting massive particle) deposits10 keV. By how much does the temperature rise?(The specific heat of argon at 1.1 K is c sp = 8 ×10 −5 J/(g K).)6. Derive (4.13) using four-momenta.7. Work out the maximum energy which can be transferredto an electron in a Compton process! As an example usethe photon transition energy of 662 keV emitted by anexcited 137 Ba nucleus after a beta decay from 137 Cs,137 Cs → 137 Ba ∗ + e − +¯ν e✲ 137 Ba + γ(662 keV) .What kind of energy does the electron get for infinitelylarge photon energies? Is there, on the other hand, a minimumenergy for the backscattered photon in this limit?8. Figure 4.2 shows the energy loss of charged particlesas given by the Bethe–Bloch formula. The abscissa isgiven as momentum and also as product of the normalizedvelocity β and the Lorentz factor γ . Show thatβγ = p/m 0 c holds.9. Equation (4.11) shows that the number of emitted Cherenkovphotons N is proportional to 1/λ 2 .Thewavelengthfor X-ray photons is shorter than that for the visiblelight region. Why then is Cherenkov light not emittedin the X-ray region?