PENELOPE 2003 - OECD Nuclear Energy Agency

3.2. Inelastic collisions 81
where PMW,I(ξ) −1 is the inverse of the cumulative distribution function, which is given by
⎧
ξA
(1 − B)(1 + A) − ξ
⎪⎨
if 0 ≤ ξ < ξ 0 ,
PMW,I(ξ) −1 = 〈µ〉 if ξ 0 ≤ ξ < ξ 0 + B, (3.33)
with
⎪⎩
(ξ − B)A
(1 − B)(1 + A) − (ξ − B)
ξ 0 = (1 − B)
if ξ 0 + B ≤ ξ ≤ 1,
(1 + A)〈µ〉
. (3.34)
A + 〈µ〉
To sample µ in the restricted interval (µ c ,1), we can still use the inverse transform
method, eq. (3.32), but with the random number ξ sampled uniformly in the interval
(ξ c ,1) with
ξ c = P MW,I (µ c ). (3.35)
• Case II. The cumulative distribution function is
P MW,II (µ) ≡
=
∫ µ
0
⎧
⎪⎨
⎪⎩
p MW,II (µ ′ ) dµ ′
(1 + A)µ
(1 − B)
A + µ
(1 + A)µ
(1 − B)
[µ
A + µ + B 4 2 − µ + 1 ]
4
if 0 ≤ µ < 1 2 ,
if 1 2 ≤ µ ≤ 1. (3.36)
In principle, to sample µ in (0,1), we can adopt the inverse transform method. The
sampling equation
ξ = P MW,II (µ) (3.37)
can be cast in the form of a cubic equation. This equation can be solved either by using
the analytical solution formulas for the cubic equation, which are somewhat complicated,
or numerically, e.g. by the Newton-Raphson method. We employ this last procedure
to determine the cutoff deflection (see section 4.1) for mixed simulation. To sample µ
in the restricted interval (µ c ,1) we use the composition method, which is easier than
solving eq. (3.37). Notice that the sampling from the (restricted) Wentzel and from the
triangle distributions can be performed analytically by the inverse transform method.
3.2 Inelastic collisions
The dominant energy loss mechanisms for electrons and positrons with intermediate and
low energies are inelastic collisions, i.e. interactions that produce electronic excitations
and ionizations in the medium. The quantum theory of inelastic collisions of charged
particles with individual atoms and molecules was first formulated by Bethe (1930, 1932)