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 (๐‘‘๐ธ๐‘‘๐‘ฅ)๐‘–๐‘œ๐‘›=22.6 MeVcm, in lead (๐‘‘๐ธ๐‘‘๐‘ฅ)๐‘–๐‘œ๐‘›=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.


โˆ’๐‘‘๐ธโ„๐‘‘๐‘ฅ=3.1โˆ—105(๐‘z2ฯโ„A๐›ฝ2)(11.2+๐‘™๐‘›(๐›ฝ2โ„๐‘(1โˆ’๐›ฝ2))โˆ’๐›ฝ2) [๐‘’๐‘‰โ„๐‘๐‘š]

Z – nuclear charge, z – particle charge, A – mass number, ฯ – substance density, ๐›ฝ=Vโ„c.

Total particle energy: ๐ธ=๐ธ๐‘˜+๐‘š๐‘2=๐‘š๐‘2โ„โˆš1โˆ’๐›ฝ2, ๐›ฝ2=๐›ผ2+2๐›ผโ„๐›ผ2+2๐›ผ+1 (๐›ผ=๐ธ๐‘˜๐‘š๐‘2)


1) ๐›ฝ2=0.539, ๐‘‘๐ธโ„๐‘‘๐‘ฅ=6.2 MeVโ„cm, 2) ๐›ฝ2=0.736, ๐‘‘๐ธโ„๐‘‘๐‘ฅ=5.1 MeVโ„cm, 3) ๐›ฝ2=0.97, ๐‘‘๐ธโ„๐‘‘๐‘ฅ=4.8 MeVโ„cm)

Task #3

Calculate the threshold proton energy for the photoproduction of ฯ€0 mesons in the interaction of a proton with a photon of CMB (cosmic microwave background) p+ฮณ โ€“> p+ฯ€0


mฯ€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 ฯ„(Al)=24 (cm2โ„g) for Eฮณ=10 KeV

ฯ„(Al)[cm-1] = ฯ„(Al)[ cm2โ„g]ร—ฯ[gโ„cm3] = 24.3ร—2.7 = 65.6 cm-1

๐ผโ„๐ผ0=๐‘’โˆ’ฯ„๐‘ฅ=99%=0.99, ๐‘ฅโ‰คโˆ’1โ„ฯ„ln(๐ผโ„๐ผ0)


โ‰ค 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 137โ„๐‘1โ„3 < ๐ธโ„๐‘š๐‘2 (58<98), we use:

(๐‘‘๐ธโ„๐‘‘๐‘ฅ)๐‘Ÿ๐‘Ž๐‘‘ = โˆ’p๐‘๐ดโ„๐ด๐ธ ๐‘2๐‘Ÿ02โ„137 (4๐‘™๐‘›(183โ„๐‘1โ„3)+2โ„9) [๐‘’๐‘‰โ„๐‘๐‘š]

If ๐ธ < ๐‘š๐‘2, we use:

(๐‘‘๐ธโ„๐‘‘๐‘ฅ)๐‘Ÿ๐‘Ž๐‘‘ = โˆ’16โ„3(p๐‘๐ดโ„๐ด)๐ธ ๐‘๐‘Ÿ02โ„137 [๐‘’๐‘‰โ„๐‘๐‘š]

If ๐ธโ„๐‘š๐‘2 < 137โ„๐‘1โ„3, we use:

(๐‘‘๐ธโ„๐‘‘๐‘ฅ)๐‘Ÿ๐‘Ž๐‘‘ = โˆ’p๐‘๐ดโ„๐ด๐ธ ๐‘2๐‘Ÿ02โ„137 (4๐‘™๐‘›(2Eโ„mec2)+4โ„3) [๐‘’๐‘‰โ„๐‘๐‘š]

(๐‘‘๐ธโ„๐‘‘๐‘ฅ)ion = โˆ’3.1โˆ—105โˆ—๐‘z2pโ„๐ด๐›ฝ2(11.2+๐‘™๐‘›(๐›ฝ2โ„๐‘(1โˆ’๐›ฝ2))โˆ’๐›ฝ2)


(๐‘‘๐ธโ„๐‘‘๐‘ฅ)๐‘Ÿ๐‘Ž๐‘‘ = โˆ’5.2 [M๐‘’๐‘‰โ„๐‘๐‘š]

(๐‘‘๐ธโ„๐‘‘๐‘ฅ)ion = โˆ’6 [M๐‘’๐‘‰โ„๐‘๐‘š]

(๐‘‘๐ธโ„๐‘‘๐‘ฅ)๐‘Ÿ๐‘Ž๐‘‘/(๐‘‘๐ธโ„๐‘‘๐‘ฅ)ion ≈ 1.2

Tasks for independent solving

Task #6

Calculate the threshold electron energy for the photoproduction of an electron-positron pair: e+ฮณ โ€“> 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 eVโ„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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