THE EGS5 CODE SYSTEM

second term on the right-hand side to yield
where
( ) dσ
= r2 0
dΩ bC,i 2
( ˘kc
˘k 0
) 2 ( ˘kc
˘k 0
+ ˘k 0
˘k c
− sin 2 θ
)
S IA
i (˘k 0 , θ, Z), (2.420)
∫ pi,max
Si IA (˘k 0 , θ, Z) = J i (p z )dp z . (2.421)
−∞
Here, Z is the atomic number and Si
IA (˘k 0 , θ, Z) is the called the incoherent scattering function of
the i-th shell electrons in the impulse approximation calculated by Ribberfors and Berggren[135],
and p i,max is obtained by putting ˘k = ˘k 0 − I i in Equation 2.416. Note that Si
IA (˘k 0 , θ, Z) converges
to the number of electrons in each sub-shell when p i,max → ∞. The singly-differential Compton
cross section of a whole atom is obtained by summing Equation 2.420 for all of the sub-shells,
where
( ) dσ IA
= r2 0
dΩ bC 2
( ˘kc
˘k 0
) 2 ( ˘kc
˘k 0
+ ˘k 0
˘k c
− sin 2 θ
S IA (˘k 0 , θ, Z) = ∑ i
)
S IA (˘k 0 , θ, Z), (2.422)
S IA
i (˘k 0 , θ, Z). (2.423)
Here, S IA (˘k 0 , θ, Z) is the incoherent scattering function of the atom in the impulse approximation.
Note that an alternative computation of the incoherent scattering function based on Waller-Hartree
theory[178] and denoted as S W H (x, Z) has been widely used in modeling electron binding effects
on the angular distribution of Compton scattered photons. In this representation of the incoherent
scattering function, x is the momentum transfer in Å, given by
x = ˘k ( )
0 (keV) θ
12.399 sin , (2.424)
2
and equivalent to q of Equation 2.409. Using S W H (x, Z) as the incoherent scattering function, the
differential Compton scattering cross section is given by
( ) ( )
dσ
W H
2 (
= r2 0 ˘kc ˘kc
+ ˘k
)
0
− sin
dΩ bC 2 ˘k 0
˘k 0
˘k 2 θ S W H (x, Z), (2.425)
c
which is the simply the Klein-Nishina cross section from before multiplied by the incoherent scattering
function. Close agreement between S IA (˘k 0 , θ, Z) and S W H (x, Z) for several atoms has been
shown by Ribberfors [135], though Namito et al.[118] have pointed out differences between S IA and
S W H at low energies. As noted earlier, as S W H (x, Z) increases from a value of 0 at x=0 to Z as
x → ∞, the net effect of atomic binding as defined through the incoherent scattering function is
to decrease the Klein-Nishina cross section in the forward direction for low energies, especially for
high Z materials.
The total bound Compton scattering cross section of an atom can be obtained by integrating
Equation 2.425 over the solid angle (Ω),
σ W H
bC
∫ 4π ( ) dσ W H
= dΩ. (2.426)
dΩ bC
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