PENELOPE 2003 - OECD Nuclear Energy Agency

C
u
2.3. Incoherent (Compton) scattering 47
15 00
m e
c J( p z
)
1000
5 00
Au
Al
0
0.001 0.010 0.100
p z
/ m e
c
Figure 2.6: Atomic Compton profiles (p z > 0) for aluminium, copper and gold. The continuous
curves are numerical Hartree-Fock profiles tabulated by Biggs et al. (1975). The dashed
curves represent the analytical profiles defined by eq. (2.57). (Adapted from Brusa et al.,
1996.)
is assumed. In the Hartree-Fock approximation for closed-shell configurations, the momentum
distribution of the electrons in an atomic shell, obtained by adding the contributions
of the orbitals in that shell, is isotropic. For an isotropic distribution, expression
(2.39) simplifies to
J i (p z ) = 2π
The atomic Compton profile is given by
∫ ∞
|p z|
p ρ i (p) dp. (2.41)
J(p z ) = ∑ i
f i J i (p z ), (2.42)
where f i is the number of electrons in the i-th shell and J i (p z ) is the one-electron profile
of this shell. The functions J(p z ) and J i (p z ) are both bell-shaped and symmetrical about
p z = 0 (see fig. 2.6). Extensive tables of Hartree-Fock Compton profiles for the elements
have been published by Biggs et al. (1975). These numerical profiles are adequate for
bound electron shells. In the case of conductors, the one-electron Compton profile for
conduction electrons may be estimated by assuming that these form a free-electron gas
with ρ e electrons per unit volume. The one-electron profile for this system is (see e.g.
Cooper, 1971)
J feg
i (p z ) = 3 ( )
1 − p2 z
Θ(p
4p F p 2 F − |p z |),
F
J feg
i (0) = 3
4p F
, (2.43)
where p F ≡ ¯h(3π 2 ρ e ) 1/3 is the Fermi momentum. For scattering in a compound material,