PENELOPE 2003 - OECD Nuclear Energy Agency

2.1. Coherent (Rayleigh) scattering 37
where
dσ T (θ)
dΩ
= 1 + cos 2 θ
r2 e
(2.3)
2
is the classical Thomson DCS for scattering by a free electron at rest, θ is the polar
scattering angle (see fig. 2.1) and F (q, Z) is the atomic form factor. The quantity r e is
the classical electron radius and q is the magnitude of the momentum transfer given by
q = 2(E/c) sin(θ/2) = (E/c) [2(1 − cos θ)] 1/2 . (2.4)
In the literature on x-ray crystallography, the dimensionless variable
is normally used instead of q.
x ≡ q 10−8 cm
4π¯h
= 20.6074
q
m e c
(2.5)
The atomic form factor can be expressed as the Fourier transform of the atomic
electron density ρ(r) which, for a spherically symmetrical atom, simplifies to
∫ ∞
F (q, Z) = 4π ρ(r) sin(qr/¯h)
0 qr/¯h r2 dr. (2.6)
F (q, Z) is a monotonically decreasing function of q that varies from F (0, Z) = Z
to F (∞, Z) = 0. The most accurate form factors are those obtained from Hartree-
Fock or configuration-interaction atomic-structure calculations; here we adopt the nonrelativistic
atomic form factors tabulated by Hubbell et al. (1975). Although relativistic
form factors are available (Doyle and Turner, 1968), Hubbell has pointed out that the
non-relativistic form factors yield results in closer agreement with experiment (Cullen
et al., 1997).
where
with
In the calculations, we use the following analytical approximation
⎧
⎪⎨ f(x, Z) ≡ Z 1 + a 1x 2 + a 2 x 3 + a 3 x 4
F (q, Z) =
(1 + a 4 x 2 + a 5 x 4 ) 2 ,
⎪⎩ max {f(x, Z), F K (q, Z)} if Z > 10 and f(x, Z) < 2,
Q =
F K (q, Z) ≡
(2.7)
sin(2b arctan Q)
bQ (1 + Q 2 ) b , (2.8)
q
2m e ca , b = √ 1 − a 2 , a ≡ α(Z − 5/16), (2.9)
where α is the fine-structure constant. The function F K (q, Z) is the contribution to
the atomic form factor due to the two K-shell electrons (see e.g. Baró et al., 1994a).
The parameters of expression f(x, Z) for Z = 1 to 92, which have been determined by
Baró et al. (1994a) by numerically fitting the atomic form factors tabulated by Hubbell
et al. (1975), are included in the block data subprogram PENDAT. The average relative