PENELOPE 2003 - OECD Nuclear Energy Agency

2.3. Incoherent (Compton) scattering 45
(i) Generate ν from π(ν) by using eq. (2.31).
(ii) Generate a random number ξ.
(iii) If ξg(0) > g(ν), go to step (i).
(iv) Deliver cos θ e = 1 − ν.
The efficiency of this algorithm is ∼ 0.33 at low energies and increases slowly with E e ;
for E e = 1 MeV, the efficiency is 0.4. As photoelectric absorption occurs at most once
in each photon history, this small sampling efficiency does not slow down the simulation
significantly.
2.3 Incoherent (Compton) scattering
In Compton scattering, a photon of energy E interacts with an atomic electron, which
absorbs it and re-emits a secondary (Compton) photon of energy E ′ in the direction Ω =
(θ, φ) relative to the direction of the original photon. In penelope, Compton scattering
events are described by means of the cross section obtained from the relativistic impulse
approximation (Ribberfors, 1983). Contributions from different atomic electron shells
are considered separately. After a Compton interaction with the i-th shell, the active
target electron is ejected to a free state with kinetic energy E e = E −E ′ −U i > 0, where
U i is the ionization energy of the considered shell, and the residual atom is left in an
excited state with a vacancy in the i-th shell.
In the case of scattering by free electrons at rest, the conservation of energy and momentum
implies the following relation between the energy E ′ of the scattered (Compton)
photon and the scattering angle θ [cf. eq. (A.19)]
E ′ ≡
E
1 + κ(1 − cos θ) ≡ E C, (2.32)
where κ = E/m e c 2 , as before. The DCS for Compton scattering by a free electron at
rest is given by the familiar Klein-Nishina formula,
dσ KN
Co
dΩ
= r2 e
2
( EC
E
) 2 ( EC
E + E )
− sin 2 θ . (2.33)
E C
Although this simple DCS was generally used in old Monte Carlo transport codes, it
represents only a rough approximation for the Compton interactions of photons with
atoms. In reality, atomic electrons are not at rest, but move with a certain momentum
distribution, which gives rise to the so-called Doppler broadening of the Compton line.
Moreover, transitions of bound electrons are allowed only if the energy transfer E − E ′
is larger than the ionization energy U i of the active shell (binding effect).
The impulse approximation accounts for Doppler broadening and binding effects
in a natural, and relatively simple, way. The DCS is obtained by considering that