PENELOPE 2003 - OECD Nuclear Energy Agency

80 Chapter 3. Electron and positron interactions
chemical binding effects) in Monte Carlo simulation at these energies is justified.
1Ε−15
dσ el
/dΩ (cm 2 /sr)
1Ε−16
1Ε−17
1Ε−18
1 0 0 eV
2 . 5 k eV
H 2
O
c o h er en t
i n c o h er en t
1Ε−19
1Ε−20
0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0
θ (deg)
Figure 3.6: DCSs for elastic scattering of electrons by water molecules, calculated as the
coherent sum of scattered waves, eq. (3.30), and from the additivity approximation (incoherent
sum).
3.1.2 Simulation of single elastic events with the MW model
As mentioned above, the angular distribution in single elastic events is axially symmetrical
about the direction of incidence. Hence, the azimuthal scattering angle φ is sampled
uniformly in the interval (0, 2π) using the sampling formula φ = 2πξ. In detailed simulations,
µ is sampled in the whole interval (0,1). However, we shall also make use of
the MW model for mixed simulation (see chapter 4), in which only hard events, with
deflection µ larger than a given cutoff value µ c , are sampled individually. In this section
we describe analytical (i.e. exact) methods for random sampling of µ in the restricted
interval (µ c , 1).
• Case I. The cumulative distribution function of p MW,I (µ) is
P MW,I (µ) ≡
∫ µ
0
⎧
⎪⎨
p MW,I (µ ′ ) dµ ′ =
⎪⎩
(1 + A)µ
(1 − B)
A + µ
(1 + A)µ
B + (1 − B)
A + µ
if 0 ≤ µ < 〈µ〉,
if 〈µ〉 ≤ µ ≤ 1.
(3.31)
Owing to the analytical simplicity of this function, the random sampling of µ can be
performed by using the inverse transform method (section 1.2.2). The sampling equation
for µ in (0,1) reads
µ = PMW,I −1 (ξ), (3.32)