Tasks with hints and answers

Task #1

Estimate the total specific energy losses of electrons with an energy of 150 MeV in aluminum and lead.




In aluminum (𝑑𝐸𝑑π‘₯)π‘–π‘œπ‘›=22.6 MeVcm, in lead (𝑑𝐸𝑑π‘₯)π‘–π‘œπ‘›=310 MeVcm

Task #2

Determine the specific ionization loss of muons in aluminum if their kinetic energy is: 1) 50 MeV, 2) 100 MeV, 3) 500 MeV.


βˆ’π‘‘πΈβ„π‘‘π‘₯=3.1βˆ—105(𝑍z2ρ⁄A𝛽2)(11.2+𝑙𝑛(𝛽2⁄𝑍(1βˆ’π›½2))βˆ’π›½2) [π‘’π‘‰β„π‘π‘š]

Z – nuclear charge, z – particle charge, A – mass number, ρ – substance density, 𝛽=V⁄c.

Total particle energy: 𝐸=πΈπ‘˜+π‘šπ‘2=π‘šπ‘2β„βˆš1βˆ’π›½2, 𝛽2=𝛼2+2𝛼⁄𝛼2+2𝛼+1 (𝛼=πΈπ‘˜π‘šπ‘2)


1) 𝛽2=0.539, 𝑑𝐸⁄𝑑π‘₯=6.2 MeV⁄cm, 2) 𝛽2=0.736, 𝑑𝐸⁄𝑑π‘₯=5.1 MeV⁄cm, 3) 𝛽2=0.97, 𝑑𝐸⁄𝑑π‘₯=4.8 MeV⁄cm)

Task #3

Calculate the threshold proton energy for the photoproduction of Ο€0 mesons in the interaction of a proton with a photon of CMB (cosmic microwave background) p+Ξ³ –> p+Ο€0


mΟ€0 = 134.98 MeV, mp = 938.27 MeV.



144.7 MeV

Task #4

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?


Linear absorption coefficient is Ο„(Al)=24 (cm2⁄g) for EΞ³=10 KeV

Ο„(Al)[cm-1] = Ο„(Al)[ cm2⁄g]×ρ[g⁄cm3] = 24.3Γ—2.7 = 65.6 cm-1

𝐼⁄𝐼0=π‘’βˆ’Ο„π‘₯=99%=0.99, π‘₯β‰€βˆ’1⁄τln(𝐼⁄𝐼0)


≀ 0.0015 mm

Task #5

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 < πΈβ„π‘šπ‘2 (58<98), we use:

(𝑑𝐸⁄𝑑π‘₯)π‘Ÿπ‘Žπ‘‘ = βˆ’p𝑁𝐴⁄𝐴𝐸 𝑍2π‘Ÿ02⁄137 (4𝑙𝑛(183⁄𝑍1⁄3)+2⁄9) [π‘’π‘‰β„π‘π‘š]

If 𝐸 < π‘šπ‘2, we use:

(𝑑𝐸⁄𝑑π‘₯)π‘Ÿπ‘Žπ‘‘ = βˆ’16⁄3(p𝑁𝐴⁄𝐴)𝐸 π‘π‘Ÿ02⁄137 [π‘’π‘‰β„π‘π‘š]

If πΈβ„π‘šπ‘2 < 137⁄𝑍1⁄3, we use:

(𝑑𝐸⁄𝑑π‘₯)π‘Ÿπ‘Žπ‘‘ = βˆ’p𝑁𝐴⁄𝐴𝐸 𝑍2π‘Ÿ02⁄137 (4𝑙𝑛(2E⁄mec2)+4⁄3) [π‘’π‘‰β„π‘π‘š]

(𝑑𝐸⁄𝑑π‘₯)ion = βˆ’3.1βˆ—105βˆ—π‘z2p⁄𝐴𝛽2(11.2+𝑙𝑛(𝛽2⁄𝑍(1βˆ’π›½2))βˆ’π›½2)


(𝑑𝐸⁄𝑑π‘₯)π‘Ÿπ‘Žπ‘‘ = βˆ’5.2 [Mπ‘’π‘‰β„π‘π‘š]

(𝑑𝐸⁄𝑑π‘₯)ion = βˆ’6 [Mπ‘’π‘‰β„π‘π‘š]

(𝑑𝐸⁄𝑑π‘₯)π‘Ÿπ‘Žπ‘‘/(𝑑𝐸⁄𝑑π‘₯)ion ≈ 1.2

Tasks for independent solving

Task #6

Calculate the threshold electron energy for the photoproduction of an electron-positron pair: e+Ξ³ –> e+e+e+


8066 MeV

Task #7

Estimate the ratio of specific ionization losses in iron for protons and electrons with energies: 1) 10 MeV, 2) 100 MeV, 3) 1 GeV.


1) 17.5, 2) 2.4, 3) 0.6

Task #8

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)]

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