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4.2.2 Attenuation of photons
The intensity I(t) of a photon beam after passing through
t kg m−2 of matter is given by
where μ/ρ is the mass absorption
coefficient. The values of μ/ρ given in the
accompanying figure and table refer to the attenuation of a collimated beam of
photons striking the attenuating material and an emerging beam consisting only
of those photons which are undeflected and have suffered no energy loss.
Inclusion of secondary products in the transmitted beam reduces this
attenuation. The values given here are taken from Hubbell (1982) Int. J.
Appl. Radioat. Isot., 33, 1269 and Hubbell, Gimm and Overbo (1980)
J. Phys. Chem. Ref. Data, 9, 1023.
The three main processes by which photons interact with
matter are photoelectric absorption which dominates at low energies, incoherent
(Compton) scattering which is most important between about 0.2 and 5 MeV, and
pair production which dominates at high energies. These three components, and
others, can be obtained separately from the two references above. The
photonuclear interaction is excluded from the values given here.
If the mass absorption coefficient
μ/ρ is written in terms of atomic cross-sections
σ as
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μ/ρ
(NA/A) ×
(σphotoelectric + σCompton +
σpair) |
where NA is Avogadro's number and
A is the atomic weight of the attenuating material, then, for purposes
of rough guidance only, approximate expressions for these component
cross-sections are as follows. The photoelectric cross-section is given roughly
by
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σphotoelectric(m2)
6 ×10−37
Z4.5/Eγ3 |
where Z is the atomic number of the attenuating material and
Eγ is the energy of
the photon in MeV. The kinematics of Compton scattering are described by
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Eγ′ =
mec2/(1 − cos θ +
mec2/Eγ) |
where Eγ is the incident photon energy,
mec2 is the electron rest energy (0.511
MeV), θ is the scattering angle and Eγ′ is the scattered photon
energy. At low energies (Eγ <<
mec2)
| |
σCompton
(8/3)πre2Z
= 0.665 × 10−28Z m2, |
and at high energies (Eγ >>
mec2)
| |
σCompton
πre2Z(1
+ 2 ln(2Eγ/(mec2)))/(2Eγ/(mec2) |
where re = 2.82 × 10−15 m is
the classical electron radius. The pair production cross-section is given
roughly by
| |
σpair
Z2re2(ln(2Eγ/(mec2))
− 2.6)/44. |
Mass absorption
coefficients for composite materials may be calculated by taking the average
value of the absorption coefficients for the constituents weighted in
proportion to their abundance by weight.
Uncertainties in the mass absorption coefficients are estimated to be
about ±5% up to 5 keV
and about ±2% above 5
keV. Above 10 MeV the photonuclear interaction can contribute an extra ~
1–5% to the mass
absorption coefficient.
Mass obsorption coefficient as a function of element
and photon energy in units of 0.01 m2 kg−1
|
MeV
|
Element |
|
H |
Be |
C |
Al |
Fe |
Ge |
Ag |
I |
W |
Pb |
U |
Con-
crete† |
Water |
|
0.2 |
2.429 |
1.089 |
1.229 |
1.223 |
1.458 |
1.658 |
2.963 |
3.650 |
7.844 |
9.985 |
12.98 |
1.27 |
1.370 |
|
0.3 |
2.112 |
0.946 |
1.066 |
1.042 |
1.098 |
1.130 |
1.557 |
1.768 |
3.238 |
4.026 |
5.191 |
1.08 |
1.187 |
|
0.4 |
1.893 |
0.847 |
0.954 |
0.928 |
0.940 |
0.933 |
1.130 |
1.215 |
1.925 |
2.323 |
2.922 |
0.96 |
1.061 |
|
0.5 |
1.729 |
0.774 |
0.871 |
0.845 |
0.841 |
0.821 |
0.931 |
0.969 |
1.378 |
1.613 |
1.976 |
0.88 |
0.969 |
|
0.6 |
1.599 |
0.716 |
0.806 |
0.780 |
0.770 |
0.745 |
0.814 |
0.830 |
1.093 |
1.248 |
1.490 |
0.81 |
0.896 |
|
0.8 |
1.405 |
0.629 |
0.708 |
0.684 |
0.670 |
0.643 |
0.676 |
0.674 |
0.806 |
0.887 |
1.016 |
0.71 |
0.787 |
|
1.0 |
1.263 |
0.565 |
0.636 |
0.615 |
0.599 |
0.573 |
0.592 |
0.584 |
0.662 |
0.710 |
0.789 |
0.64 |
0.707 |
|
1.5 |
1.027 |
0.460 |
0.518 |
0.501 |
0.488 |
0.466 |
0.475 |
0.465 |
0.500 |
0.522 |
0.559 |
0.52 |
0.576 |
|
2.0 |
0.877 |
0.394 |
0.444 |
0.432 |
0.426 |
0.409 |
0.421 |
0.412 |
0.443 |
0.461 |
0.488 |
0.45 |
0.494 |
|
3.0 |
0.692 |
0.314 |
0.356 |
0.354 |
0.362 |
0.352 |
0.375 |
0.372 |
0.408 |
0.423 |
0.445 |
0.37 |
0.397 |
|
4.0 |
0.581 |
0.266 |
0.305 |
0.311 |
0.331 |
0.328 |
0.361 |
0.361 |
0.404 |
0.420 |
0.439 |
0.32 |
0.340 |
|
5.0 |
0.505 |
0.235 |
0.271 |
0.284 |
0.315 |
0.316 |
0.358 |
0.361 |
0.410 |
0.427 |
0.446 |
0.29 |
0.303 |
|
6.0 |
0.450 |
0.212 |
0.247 |
0.265 |
0.306 |
0.311 |
0.360 |
0.366 |
0.421 |
0.439 |
0.458 |
0.27 |
0.277 |
|
8.0 |
0.375 |
0.182 |
0.215 |
0.244 |
0.299 |
0.310 |
0.372 |
0.382 |
0.447 |
0.468 |
0.488 |
0.24 |
0.243 |
|
10.0 |
0.325 |
0.163 |
0.196 |
0.232 |
0.299 |
0.316 |
0.388 |
0.400 |
0.475 |
0.497 |
0.519 |
0.23 |
0.222 |
|
15.0 |
0.254 |
0.136 |
0.170 |
0.220 |
0.309 |
0.334 |
0.428 |
0.446 |
0.538 |
0.566 |
0.593 |
0.22 |
0.194 |
|
20.0 |
0.215 |
0.123 |
0.158 |
0.217 |
0.322 |
0.353 |
0.461 |
0.482 |
0.589 |
0.620 |
0.651 |
0.21 |
0.181 |
|
30.0 |
0.175 |
0.110 |
0.147 |
0.220 |
0.347 |
0.385 |
0.513 |
0.540 |
0.665 |
0.702 |
0.739 |
0.21 |
0.171 |
|
40.0 |
0.154 |
0.104 |
0.144 |
0.225 |
0.367 |
0.410 |
0.551 |
0.581 |
0.720 |
0.761 |
0.802 |
0.21 |
0.168 |
|
50.0 |
0.142 |
0.102 |
0.143 |
0.231 |
0.383 |
0.430 |
0.581 |
0.613 |
0.762 |
0.806 |
0.849 |
0.22 |
0.167 |
|
60.0 |
0.134 |
0.100 |
0.143 |
0.236 |
0.396 |
0.446 |
0.604 |
0.638 |
0.795 |
0.841 |
0.887 |
0.22 |
0.168 |
|
80.0 |
0.124 |
0.099 |
0.144 |
0.245 |
0.417 |
0.471 |
0.640 |
0.676 |
0.844 |
0.893 |
0.943 |
0.23 |
0.170 |
|
100.0 |
0.119 |
0.099 |
0.146 |
0.252 |
0.433 |
0.489 |
0.666 |
0.704 |
0.880 |
0.931 |
0.983 |
0.24 |
0.173 |
|
150.0 |
0.114 |
0.101 |
0.150 |
0.264 |
0.459 |
0.519 |
0.709 |
0.749 |
0.936 |
0.991 |
1.047 |
0.25 |
0.178 |
|
200.0 |
0.112 |
0.102 |
0.154 |
0.272 |
0.476 |
0.538 |
0.734 |
0.777 |
0.970 |
1.027 |
1.085 |
0.25 |
0.183 |
|
300.0 |
0.112 |
0.105 |
0.159 |
0.282 |
0.495 |
0.559 |
0.765 |
0.809 |
1.010 |
1.069 |
1.131 |
0.26 |
0.189 |
|
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†Approximate composition by weight of
concrete: 50% oxygen, 31% silicon, 8% calcium, 5% aluminium, 6% other materials.
(Click the Image to View Larger Image)
Mass absorption coefficient (m/r) as a function of element for a set of
photon energies between 1 and 200 keV.
D.J.S. Findlay
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