PENELOPE 2003 - OECD Nuclear Energy Agency

48 Chapter 2. Photon interactions
the molecular Compton profile is obtained as the sum of atomic profiles of the atoms in
a molecule (additivity rule).
The factor F (p z ) in eq. (2.34) is approximately given by
F (p z ) ≃ 1 + cq C
E
(
1 + E C(E C − E cos θ)
(cq C ) 2
)
pz
m e c , (2.44)
where q C is the momentum transfer associated with the energy E ′ = E C of the Compton
line,
q C ≡ 1 √
E
c
2 + EC 2 − 2EE C cos θ. (2.45)
Expression (2.44) is accurate only for small |p z |-values. For large |p z |, J i (p z ) tends to
zero and the factor F (p z ) has no effect on the DCS. We use the values given by expression
(2.44) only for |p z | < 0.2m e c and take F (±|p z |) = F (±0.2m e c) for |p z | > 0.2m e c. Owing
to the approximations introduced, negative values of F may be obtained for large |p z |;
in this case, we must set F = 0.
We can now introduce the effect of electron binding: Compton excitations are allowed
only if the target electron is promoted to a free state, i.e. if the energy transfer E − E ′
is larger than the ionization energy U i of the active shell. Therefore the atomic DCS,
including Doppler broadening and binding effects, is given by
d 2 σ Co
dE ′ dΩ = r2 e
2
( ) 2 ( EC EC
E
E + E )
− sin 2 θ
E C
× F (p z )
( ∑
i
)
f i J i (p z ) Θ(E − E ′ dpz
− U i )
dE ′, (2.46)
where Θ(x) (= 1 if x > 0, = 0 otherwise) is the Heaviside step function. In the
calculations we use the ionization energies U i given by Lederer and Shirley (1978), fig.
2.4. The DCS for scattering of 10 keV photons by aluminium atoms is displayed in fig.
2.7, for θ = 60 and 180 degrees, as a function of the fractional energy of the emerging
photon. The DCS for a given scattering angle has a maximum at E ′ = E C ; its shape
resembles that of the atomic Compton profile, except for the occurrence of edges at
E ′ = E − U i .
In the case of scattering by free electrons at rest we have U i = 0 (no binding) and
J i (p z ) = δ(p z ) (no Doppler broadening). Moreover, from eq. (2.37) E ′ = E C , so that
photons scattered through an angle θ have energy E C . Integration of the DCS, eq.
(2.46), over E ′ then yields the familiar Klein-Nishina cross section,
dσ KN
Co
dΩ
= Z r2 e
2
( EC
E
) 2 ( EC
E + E )
− sin 2 θ , (2.47)
E C
for the Z atomic electrons [cf. eq. (2.33)]. For energies of the order of a few MeV and
larger, Doppler broadening and binding effects are relatively small and the free-electron
theory yields results practically equivalent to those of the impulse approximation.