PENELOPE 2003 - OECD Nuclear Energy Agency

2.4. Electron-positron pair production 61
that the functions φ 1 (ɛ) and φ 2 (ɛ) are non-negative and attain their maximum values at
ɛ = 1/2.
4 0
Pb
2 00 M e V
d σ pp
/d ε (barn)
3 0
2 0
50
2 0
1 0
5
1 0
2 .5
1 .5
0
0.0 0.2 0.4 0.6 0.8 1 .0
ε
Figure 2.9: Pair production DCS in lead as a function of the electron reduced energy, ɛ =
(E − + m e c 2 )/E. (Adapted from Baró et al., 1994a.)
Except for a normalization constant, the PDF (2.94) can be written in the form
p pp (ɛ) = u 1 U 1 (ɛ)π 1 (ɛ) + u 2 U 2 (ɛ)π 2 (ɛ) (2.95)
with
) 2
φ 1 (1/2), u 2 = φ 2 (1/2), (2.96)
u 1 = 2 3
( 1
2 − 1 κ
( 1
2 − 1 κ
) −3 ( 1
2 − ɛ ) 2
, π 2 (ɛ) = 1 2
( 1
2 − 1 κ
) −1
(2.97)
π 1 (ɛ) = 3 2
and
U 1 (ɛ) = φ 1 (ɛ)/φ 1 (1/2), U 2 (ɛ) = φ 2 (ɛ)/φ 2 (1/2). (2.98)
The functions π i (ɛ) are normalized PDFs in the interval (κ −1 ,1 − κ −1 ), from which
random values of ɛ can be easily sampled by using the inverse transform method. In this
interval, the functions U i (ɛ) are positive and less than unity, i.e. they are valid rejection
functions. The generation of random values of ɛ from the distribution (2.95) can now
be performed by combining the composition and rejection methods (see section 1.2)
according to the following algorithm:
(i) Sample a value of the integer i (=1, 2) according to the point probabilities
p(1) = u 1
u 1 + u 2
and p(2) = u 2
u 1 + u 2
. (2.99)