PENELOPE 2003 - OECD Nuclear Energy Agency

2.3. Incoherent (Compton) scattering 49
σ Co
/dE' dΩ ( barn /sr)
2
1
0
2
Al, E= 1 0 k e V
θ = 6 0 ο
θ = 18 0 ο
(E /Z ) d
2
1
0
0.90 0.95 1.00
E'/E
Figure 2.7: DCS for Compton scattering of 10 keV photons by aluminium atoms at the
indicated scattering angles. The continuous curves represent the DCS (2.46) calculated using
the Hartree-Fock Compton profile (Biggs et al., 1975). The dashed curves are results from eq.
(2.46) with the analytical profiles given by eq. (2.57). (Adapted from Brusa et al., 1996.)
The angular distribution of scattered photons is given by the directional DCS,
∫
dσ Co d 2
dΩ = σ Co
dE ′ dΩ dE′ = r2 e
2
( EC
E
) 2 ( EC
E + E )
− sin 2 θ
E C
× ∑ i
f i Θ(E − U i )
∫ pi,max
−∞
F (p z )J i (p z ) dp z , (2.48)
where p i,max is the highest p z -value for which an electron in the i-th shell can be excited.
It is obtained from eq. (2.36) by setting E ′ = E − U i ,
p i,max (E, θ) = E(E − U i)(1 − cos θ) − m e c 2 U i
√
. (2.49)
c 2E(E − U i )(1 − cos θ) + Ui
2
Except for energies just above the shell ionization threshold, the function F (p z ) in the
integral can be replaced by unity, since p z J i (p z ) is an odd function and its integral is
close to zero, i.e. ∫ pi,max
F (p z )J i (p z ) dp z ≃ n i (p i,max ), (2.50)
−∞