PENELOPE 2003 - OECD Nuclear Energy Agency

62 Chapter 2. Photon interactions
(ii) Sample ɛ from π i (ɛ) using the sampling formulae (inverse transform method, see
section 1.2.2)
⎧ (
1 1
⎪⎨ 2 + 2 − 1 )
(2ξ − 1) 1/3 if i = 1,
κ
ɛ = (
1 1 ⎪⎩
κ + 2 − 1 )
(2.100)
2ξ if i = 2.
κ
(iii) Generate a new random number ξ.
(iv) If ξ > U i (ɛ), go to step (i).
(v) Deliver ɛ.
Notice that the quantity 2ξ − 1 may be negative and, therefore, taking its cube root
will lead to a computer error; provision of this fact must be made when programming
the algorithm. The efficiency of the algorithm is greater than 70% for energies near the
threshold, and increases with increasing photon energies. For E = 1 GeV it is of the
order of 95% for all the elements in the periodic table.
Angular distribution of the produced particles
Actually, the complete DCS for pair production is a function of the directions of the
pair of particles. As the final state involves three bodies (the nucleus and the produced
pair), the directions of the produced particles cannot be obtained from only their kinetic
energies. The polar angles of the directions of movement of the electron and positron
(θ − and θ + , fig. 2.1) relative to the direction of the incident photon are sampled from the
leading term of the expression obtained from high-energy theory (Heitler, 1954; Motz
et al., 1969)
p(cos θ ± ) = a (1 − β ± cos θ ± ) −2 , (2.101)
where a is a normalization constant and
√
E ± (E ± + 2m e c 2 )
β ± =
(2.102)
E ± + m e c 2
is the particle velocity in units of the speed of light. Random values of cos θ ± are
obtained by using the inverse transform method (see section 1.2.2), which leads to the
sampling formula
cos θ ± = 2ξ − 1 + β ±
(2ξ − 1)β ± + 1 . (2.103)
As the directions of the produced particles and the incident photon are not necessarily
coplanar, the azimuthal angles φ − and φ + of the electron and the positron are sampled
independently and uniformly in the interval (0, 2π).
It is worth stressing the fact that the produced charged particles have ranges that
are much smaller than the mean free path of the photons. Moreover, the charged
particles immediately enter a multiple elastic scattering process which randomizes their