PENELOPE 2003 - OECD Nuclear Energy Agency

2.3. Incoherent (Compton) scattering 51
0.8
4
0.6
(E /Z ) dσ Co
/dE' ( barn)
2
0
4
2
A l , E= 5 0 k e V
0.4
0.2
0.0
0.8
0.6
0.4
A l , E= 5 00 k e V
A u , E= 5 0 k e V
0.2
A u , E= 5 00 k e V
0
0.7 0.8 0.9 1 .0
E'/E
0.0
0.2 0.4 0.6 0.8 1 .0
E'/E
Figure 2.8: Energy DCSs for Compton scattering of 50 and 500 keV photons by aluminium
and gold atoms. The continuous curves represent the DCS (2.55), computed using the analytical
Compton profiles (2.57). The dashed curves are obtained from the Klein-Nishina formula
(2.56), i.e. assuming that the atomic electrons are free and at rest.
2.3.1 Analytical Compton profiles
In order to minimize the required numerical information and to simplify the random
sampling, we use approximate one-electron profiles of the form
Ji A nd ( 2
(p z ) = J i,0 d1 + d 2 J i,0 |p z | ) n−1 [
exp d
n
2
1 − ( d 1 + d 2 J i,0 |p z | ) n ] (2.57)
with
( )
√
n − 1
1/n
1
n = 2, d 1 =
=
n 2 , d 2 = 2 n d 1 1−n = √ 2.
The quantity J i,0 ≡ J i (0) is the value of the profile at p z = 0 obtained from the Hartree-
Fock orbital (Biggs et al., 1975). J i (0) is tabulated in the file PDATCONF.TAB for all
shells of the elements Z = 1 to 92. Notice that Ji A (p z ) is normalized according to eq.
(2.40). With the profiles (2.57),
[
1
∫
n A i (p pz
⎧⎪ ⎨ exp d
z) ≡ Ji A (p′ z ) dp′ z = 2 1 2 − ( ) ] 2
d 1 − d 2 J i,0 p z if p z < 0,
[
−∞
⎪ ⎩ 1 − 1 exp d 2 2 1 − ( ) ]
(2.58)
2
d 1 + d 2 J i,0 p z if p z > 0.