PENELOPE 2003 - OECD Nuclear Energy Agency

2.4. Electron-positron pair production 55
electron shell i is sampled with relative probability f i and binding effects are accounted
for by simply rejecting E ′ -values such that E − E ′ < U i .
The azimuthal scattering angle φ of the photon is sampled uniformly in the interval
(0, 2π). We assume that the Compton electron is emitted with energy E e = E − E ′ − U i
in the direction of the momentum transfer vector q = ¯hk − ¯hk ′ , with polar angle θ e and
azimuthal angle φ e = φ + π, relative to the direction of the incident photon. cos θ e is
given by
E − E ′ cos θ
cos θ e = √
E2 + E ′2 − 2EE ′ cos θ . (2.75)
When E ′ = E C , this expression simplifies to
cos θ e = E + m ec 2 (
)
E − E 1/2
C
, (2.76)
E 2m e c 2 + E − E C
which coincides with the result (A.20). Since the active electron shell is known, characteristic
x rays and Auger electrons emitted in the de-excitation of the ionized atom
can also be followed. This is important, for instance, to account for escape peaks in
scintillation or solid state detectors
Table 2.1: Average number n r of random numbers ξ needed to simulate a single incoherent
scattering event for photons with energy E in aluminium, silver and gold.
E (eV) Al Ag Au
10 3 16.6 11.9 13.4
10 4 11.0 11.4 11.5
10 5 9.5 9.8 10.0
10 6 8.2 8.2 8.3
10 7 7.5 7.5 7.5
As a measure of the efficiency of the sampling algorithm, we may consider the average
number n r of random numbers ξ required to simulate an incoherent scattering event.
n r is practically independent of the atomic number and decreases with photon energy
(see table 2.1). The increase of n r at low energies stems from the loss of efficiency
of the algorithm used to sample cos θ. Although the simulation of incoherent events
becomes more laborious as the photon energy decreases, this has only a small influence
on the speed of practical photon transport simulations since low-energy photons interact
predominantly via photoelectric absorption (see fig. 2.10 below).
2.4 Electron-positron pair production
Electron-positron pairs can be created by absorption of a photon in the vicinity of a
massive particle, a nucleus or an electron, which absorbs energy and momentum so that