PENELOPE 2003 - OECD Nuclear Energy Agency

56 Chapter 2. Photon interactions
these two quantities are conserved. The threshold energy for pair production in the field
of a nucleus (assumed of infinite mass) is 2m e c 2 . When pair production occurs in the
field of an electron, the target electron recoils after the event with appreciable kinetic
energy; the process is known as “triplet production” because it causes three visible
tracks when observed, e.g. in a cloud chamber. If the target electron is at rest, triplet
production is only possible for photons with energy larger than 4m e c 2 .
For the simulation of pair production events in the field of an atom of atomic number
Z, we shall use the following semiempirical model (Baró et al., 1994a). Our starting
point is the high-energy DCS for arbitrary screening, which was derived by Bethe and
Heitler from the Born approximation (Motz et al., 1969; Tsai, 1974). The Bethe-Heitler
DCS for a photon of energy E to create an electron-positron pair, in which the electron
has a kinetic energy E − = ɛE − m e c 2 , can be expressed as (Tsai, 1974)
dσ (BH)
pp
dɛ
{ [ɛ
= reαZ[Z 2 + η] 2 + (1 − ɛ) 2] (Φ 1 − 4f C ) + 2 }
3 ɛ(1 − ɛ)(Φ 2 − 4f C ) . (2.77)
Notice that the “reduced energy” ɛ = (E − + m e c 2 )/E is the fraction of the photon
energy that is taken away by the electron. The screening functions Φ 1 and Φ 2 are
given by integrals that involve the atomic form factor and, therefore, must be computed
numerically when a realistic form factor is adopted (e.g. the analytical one described in
section 2.1). To obtain approximate analytical expressions for these functions, we shall
assume that the Coulomb field of the nucleus is exponentially screened by the atomic
electrons (Schiff, 1968; Tsai, 1974), i.e. the electrostatic potential of the atom is assumed
to be (Wentzel model)
ϕ W (r) = Ze exp(−r/R), (2.78)
r
with the screening radius R considered as an adjustable parameter (see below). The
corresponding atomic electron density is obtained from Poisson’s equation,
ρ W (r) = 1
4πe ∇2 ϕ(r) = 1 1
4πe r
and the atomic form factor is
F W (q, Z) = 4π
∫ ∞
0
d 2
dr 2 [rϕ(r)] =
ρ W (r) sin(qr/¯h)
qr/¯h r2 dr =
Z exp(−r/R), (2.79)
4πR 2 r
Z
1 + (Rq/¯h) 2.
(2.80)
The screening functions for this particular form factor take the following analytical
expressions (Tsai, 1974)
Φ 1 = 2 − 2 ln(1 + b 2 ) − 4b arctan(b −1 ) + 4 ln(Rm e c/¯h)
Φ 2 = 4 3 − 2 ln(1 + b2 ) + 2b 2 [ 4 − 4b arctan(b −1 ) − 3 ln(1 + b −2 ) ]
+ 4 ln(Rm e c/¯h), (2.81)
where
b = Rm ec
¯h
1 1
2κ ɛ(1 − ɛ) . (2.82)