PENELOPE 2003 - OECD Nuclear Energy Agency

46 Chapter 2. Photon interactions
electrons in the i-th shell move with a momentum distribution ρ i (p). For an electron in
an orbital ψ i (r), ρ i (p) ≡ |ψ i (p)| 2 , where ψ i (p) is the wave function in the momentum
representation. The DCS for Compton scattering by an electron with momentum p
is derived from the Klein-Nishina formula by applying a Lorentz transformation with
velocity v equal to that of the moving target electron. The impulse approximation to
the Compton DCS (per electron) of the considered shell is obtained by averaging over
the momentum distribution ρ i (p).
After some manipulations, the Compton DCS of an electron in the i-th shell can be
expressed as [eq. (21) in Brusa et al., 1996]
d 2 σ Co,i
dE ′ dΩ = r2 e
2
( EC
E
) 2 ( EC
E + E )
− sin 2 θ F (p z ) J i (p z ) dp z
E C dE ′, (2.34)
where r e is the classical electron radius. E C is the energy of the Compton line, defined
by eq. (2.32), i.e. the energy of photons scattered in the direction θ by free electrons at
rest. The momentum transfer vector is given by q ≡ ¯hk − ¯hk ′ , where ¯hk and ¯hk ′ are
the momenta of the incident and scattered photons; its magnitude is
q = 1 c√
E2 + E ′2 − 2EE ′ cos θ. (2.35)
The quantity p z is the projection of the initial momentum p of the electron on the
direction of the scattering vector ¯hk ′ − ¯hk = −q; it is given by 2
or, equivalently,
p z ≡ − p · q
q
= EE′ (1 − cos θ) − m e c 2 (E − E ′ )
c 2 q
p z
m e c = E(E′ − E C )
E C cq
Notice that p z = 0 for E ′ = E C . Moreover,
dp z
= m (
ec E
+
dE ′ cq E C
E cos θ − E′
cq
(2.36)
. (2.37)
)
p z
. (2.38)
m e c
The function J i (p z ) in eq. (2.34) is the one-electron Compton profile of the active
shell, which is defined as
∫ ∫
J i (p z ) ≡ ρ i (p) dp x dp y , (2.39)
where ρ i (p) is the electron momentum distribution. That is, J i (p z ) dp z gives the probability
that the component of the electron momentum in the z-direction is in the interval
(p z , p z + dp z ). Notice that the normalization
∫ ∞
J i (p z ) dp z = 1 (2.40)
−∞
2 The expression (2.36) contains an approximation, the exact relation is obtained by replacing the
electron rest energy m e c 2 in the numerator by the electron initial total energy, √ (m e c 2 ) 2 + (cp) 2 .