THE EGS5 CODE SYSTEM

are reasonable, but at lower initial energies, atomic electron binding has the effect of decreasing
the Compton scattering cross section given by Equation 2.170, particularly in the forward direction.
Additionally, because bound atomic electrons are in motion, they emerge from Compton
interactions not at energies wholly defined by the scattering angle as given in Equation 2.173, but
with a distribution of possible energies (this effect is usually referred to as “Doppler broadening.”)
Treatments of both atomic binding effects and Doppler broadening were introduced into EGS4 by
Namito and co-workers [119, 117], and all of those methods have been incorporated into the default
version of EGS5, as options initiated through flags specified by the user.
To examine the effects of atomic binding and electron motion, we start with a more generalized
treatment of photon scattering than that of Klein and Nishina. Ribberfors derived a doubly
differential Compton scattering cross section for unpolarized photons impingent on bound atomic
electrons using the relativistic impulse approximation [134]. His result can be expressed as
where
(
d 2 σ
dΩd˘k
)
bC,i
= r2 0
2
( ) ˘kc˘k dpz
˘k 2 0
d˘k
( ˘kc
+ ˘k
)
0
− sin
˘k 0
˘k 2 θ J i (p z ) (2.415)
c
p z = −137 ˘k 0 − ˘k − ˘k 0˘k(1 − cos θ)/m
¯hc| ⃗˘k0 − ⃗˘k| , (2.416)
dp z
d˘k
=
137˘k 0
− p z(˘k − ˘k 0 cos θ)
¯hc| ⃗˘k0 − ⃗˘k|˘kc (¯hc) 2 | ⃗˘k0 − ⃗˘k| , (2.417)
2
and
˘k c =
˘k 0
1 + ˘k (2.418)
0
m
(1 − cos θ),
¯hc| ⃗˘k0 − ⃗˘k| =
√˘k2 0 + ˘k 2 − 2˘k 0˘k cos θ. (2.419)
Here, the subscript “bC ” denotes Compton scattering by a bound electron; subscript “i ” denotes
the sub-shell number corresponding to the (n, l, m)-th sub-shells; r 0 is the classical electron radius
as before; ˘k0 and ˘k are the incident and scattered photon energies, respectively, and ˘k c is the
Compton scattered photon energy for an electron at rest (Equation 2.173); p z is the projection
of the electron pre-collision momentum on the photon scattering vector in atomic units; J i (p z ) is
the Compton profile of the i-th sub-shell[35]; θ is the scattering polar angle; and m is the electron
rest mass. Note that as we are dealing with bound electrons, it is implicit in the above that the
cross section given by Equation 2.415 is 0 when ˘k > ˘k 0 − I i , where I i is the binding energy of an
electron in the i-th shell. Note also that by substituting Equation 2.417 into Equation 2.415 after
eliminating the second term on the right-hand side of Equation 2.417, one obtains an equivalent
formula to Ribberfors’ Equation 3 [134].
The singly-differential Compton cross section (in solid angle) for the scattering from a bound
electron is obtained by integrating Equation 2.415 over ˘k with the assumption that ˘k = ˘k c in the
130