PENELOPE 2003 - OECD Nuclear Energy Agency

130 Chapter 4. Electron/positron transport mechanics
of generating the angular deflection in artificial soft events. When the result given by
eq. (4.27) is applicable, the single parameter λ (s)
el,1 completely determines the multiple
scattering distribution due to soft collisions, i.e. other details of the DCS for scattering
angles less than θ c are irrelevant. However, in actual Monte Carlo simulations, the
small-angle approximation is seldom applicable.
In most practical cases the number of hard collisions per electron track can be made
relatively large by simply using a small value of the parameter C 1 [see eq. (4.10)]. When
the number of steps is large enough, say larger than ∼ 10, it is not necessary to use the
exact distribution F (s) (s; χ) to sample the angular deflection in artificial soft collisions.
Instead, we may use a simpler distribution, F a (s; χ), with the same mean and variance,
without appreciably distorting the simulation results. This is so because the details of
the adopted distribution are washed out after a sufficiently large number of steps and
will not be seen in the simulated distributions. Notice that, within the small angle
approximation, it is necessary to keep only the proper value of the first moment to
get the correct final distributions. However, if the cutoff angle θ c is not small enough,
the angular distribution F (s) (s; χ) may become sensitive to higher-order moments of
the soft single scattering distribution. Thus, by also keeping the proper value of the
variance, the range of validity of the simulation algorithm is extended, i.e. we can speed
up the simulation by using larger values of C 1 (or of λ (h)
el ) and still obtain the correct
distributions.
We now return to the notation of section 3.1, and use the variable µ ≡ (1 − cos χ)/2
to describe angular deflections in soft scattering events. The exact first and second
moments of the multiple scattering distribution F (s) (s; µ) are
〈µ〉 (s) ≡
∫ 1
0
µF a (s; µ) dµ = 1 2
[
1 − exp(−s/λ
(s)
el,1) ] (4.28)
and
∫ 1
〈µ 2 〉 (s) ≡ µ 2 F a (s; µ) dµ = 〈µ〉 (s) − 1 [
(s)
1 − exp(−s/λ el,2
0
6
)] . (4.29)
The angular deflection in soft scattering events will be generated from a distribution
F a (s; µ), which is required to satisfy eqs. (4.28) and (4.29), but is otherwise arbitrary.
penelope uses the following,
F a (s; µ) = aU 0,b (µ) + (1 − a)U b,1 (µ), (4.30)
where U u,v (x) denotes the normalized uniform distribution in the interval (u, v),
⎧
⎪⎨ 1/(v − u) if u ≤ x ≤ v,
U u,v (x) =
⎪⎩ 0 otherwise.
(4.31)
The parameters a and b, obtained from the conditions (4.28) and (4.29), are
b = 2〈µ〉(s) − 3〈µ 2 〉 (s)
1 − 2〈µ〉 (s) , a = 1 − 2〈µ〉 (s) + b. (4.32)