Tasks with hints and answers
Estimate the total specific energy losses of electrons with an energy of 150 MeV in aluminum and lead.
Determine the specific ionization loss of muons in aluminum if their kinetic energy is: 1) 50 MeV, 2) 100 MeV, 3) 500 MeV.
Z – nuclear charge, z – particle charge, A – mass number, ρ – substance density, 𝛽=V⁄c.
Total particle energy: 𝐸=𝐸𝑘+𝑚𝑐2=𝑚𝑐2⁄√1−𝛽2, 𝛽2=𝛼2+2𝛼⁄𝛼2+2𝛼+1 (𝛼=𝐸𝑘𝑚𝑐2)
1) 𝛽2=0.539, 𝑑𝐸⁄𝑑𝑥=6.2 MeV⁄cm, 2) 𝛽2=0.736, 𝑑𝐸⁄𝑑𝑥=5.1 MeV⁄cm, 3) 𝛽2=0.97, 𝑑𝐸⁄𝑑𝑥=4.8 MeV⁄cm)
Calculate the threshold proton energy for the photoproduction of π0 mesons in the interaction of a proton with a photon of CMB (cosmic microwave background) p+γ –> p+π0
What should be the thickness of the walls of the aluminum container so that they absorb no more than 1% of gamma rays with an energy of 10 keV?
τ(Al)[cm-1] = τ(Al)[ cm2⁄g]×ρ[g⁄cm3] = 24.3×2.7 = 65.6 cm-1
Determine specific radiative losses during the passage of electrons with an energy of 50 MeV by an aluminum target and to compare them to specific losses for ionization.
If 137⁄𝑍1⁄3 < 𝐸⁄𝑚𝑐2 (58<98), we use:
(𝑑𝐸⁄𝑑𝑥)𝑟𝑎𝑑 = −p𝑁𝐴⁄𝐴𝐸 𝑍2𝑟02⁄137 (4𝑙𝑛(183⁄𝑍1⁄3)+2⁄9) [𝑒𝑉⁄𝑐𝑚]
If 𝐸 < 𝑚𝑐2, we use:
(𝑑𝐸⁄𝑑𝑥)𝑟𝑎𝑑 = −16⁄3(p𝑁𝐴⁄𝐴)𝐸 𝑍𝑟02⁄137 [𝑒𝑉⁄𝑐𝑚]
If 𝐸⁄𝑚𝑐2 < 137⁄𝑍1⁄3, we use:
(𝑑𝐸⁄𝑑𝑥)𝑟𝑎𝑑 = −p𝑁𝐴⁄𝐴𝐸 𝑍2𝑟02⁄137 (4𝑙𝑛(2E⁄mec2)+4⁄3) [𝑒𝑉⁄𝑐𝑚](𝑑𝐸⁄𝑑𝑥)ion = −3.1∗105∗𝑍z2p⁄𝐴𝛽2(11.2+𝑙𝑛(𝛽2⁄𝑍(1−𝛽2))−𝛽2)
(𝑑𝐸⁄𝑑𝑥)𝑟𝑎𝑑 = −5.2 [M𝑒𝑉⁄𝑐𝑚]
(𝑑𝐸⁄𝑑𝑥)ion = −6 [M𝑒𝑉⁄𝑐𝑚]
(𝑑𝐸⁄𝑑𝑥)𝑟𝑎𝑑/(𝑑𝐸⁄𝑑𝑥)ion ≈ 1.2
Tasks for independent solving
Calculate the threshold electron energy for the photoproduction of an electron-positron pair: e–+γ –> e–+e–+e+
Estimate the ratio of specific ionization losses in iron for protons and electrons with energies: 1) 10 MeV, 2) 100 MeV, 3) 1 GeV.
Calculate the intensity of cosmic rays with kinetic energies > 1 GeV, based on the power-law form of the energy spectrum of cosmic rays with an index of 2.7 and their total energy density of 0.5 eV⁄cm3 (assume that the particles are relativistic and make the main contribution to the total energy density)
I = 2(Ek/1GeV)-2.7 [particle/(cm2*s*GeV)]