PENELOPE 2003 - OECD Nuclear Energy Agency

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64 Chapter 2. Photon interactions
and the program xcom (Berger and Hubbell, 1987), respectively. Photoelectric cross
sections for energies different from those in the tables are calculated by linear log-log
interpolation. Total cross sections for pair production are evaluated by cubic spline
log-log interpolation of the function (1 − 2m e c 2 /E) −3 σ pp , which varies slowly with the
photon energy.
1Ε+4
1Ε+4
coherent
1Ε+3
i ncoherent
1Ε+3
hotoel ectri c
1Ε+2
p a i r p rod u ct.
1Ε+2
tota
µ /ρ (cm 2 /g)
1Ε+1
1Ε+0
H 2
O
µ /ρ (cm 2 /g)
1Ε+1
1Ε+0
Pb
1Ε− 1
1Ε− 1
1Ε− 2
1Ε− 2
1Ε− 3
1Ε− 3
1Ε+3 1Ε+4 1Ε+5 1Ε+6 1Ε+7 1Ε+8 1Ε+9
1Ε+3 1Ε+4 1Ε+5 1Ε+6 1Ε+7 1Ε+8 1Ε+9
E (eV)
E (eV)
Figure 2.10: Partial and total mass attenuation coefficients of water and lead as functions of
the photon energy.
Mean free paths for coherent and incoherent scattering are computed from the DCSs
described in sections 2.1 and 2.3. The resulting values are virtually identical to those
given by the xcom program for E greater than ∼ 50 keV. At lower energies, our mean
free paths for Compton scattering deviate from those given by xcom; these were calculated
from a different theoretical model (Hubbell et al., 1975), which neglects Doppler
broadening (see e.g. Brusa et al., 1996). The evaluation of the total atomic cross section
for these processes [see eqs. (2.10) and (2.54)] involves a numerical quadrature, which is
performed by using the function SUMGA (appendix B). Notice that for high-energy photons,
the integrand in the coherent scattering cross section, eq. (2.10), is sharply peaked
at θ = 0. In such a case, the numerical integration method is not effective. For energies
larger than ∼ Z/2 MeV, we take advantage of the asymptotic behaviour shown by eq.
(2.12) to avoid time-consuming integration. Partial and total mass attenuation coefficients
for water and lead, as representatives of low- and high-Z materials, are displayed
in fig. 2.10.