PENELOPE 2003 - OECD Nuclear Energy Agency

3.2. Inelastic collisions 99
After selecting the active oscillator, the oscillator branch (distant or close) is determined
and, finally, the variables W and Q (or cos θ) are sampled from the associated
DCS. For close collisions, Q = W and, therefore, the scattering angle is obtained directly
from the energy loss.
Hard distant interactions
In distant interactions with the k-th oscillator, W = W k . The contributions of transverse
and longitudinal interactions to the restricted cross section define the relative
probabilities of these interaction modes. If the interaction is (distant) transverse, the
angular deflection of the projectile is neglected, i.e. cos θ = 1. For distant longitudinal
collisions, the (unnormalized) PDF of Q is given by [see eq. (3.58)]
⎧
1
⎪⎨
if Q
P dk (Q) = Q [1 + Q/(2m e c 2 − < Q < W k ,
)]
⎪⎩
0 otherwise,
(3.108)
where Q − is the minimum recoil energy, eq. (A.31). Random sampling from this PDF
can be performed by the inverse transform method, which gives the sampling formula
where
{ [ ( QS
Q = Q S 1 + W )] ξ
k
−
Q } −1
S
, (3.109)
W k 2m e c 2 2m e c 2
Q S ≡
Q −
1 + Q − / (2m e c 2 ) . (3.110)
Once the energy loss and the recoil energy have been sampled, the polar scattering angle
θ is determined from eq. (A.40),
cos θ = E(E + 2m ec 2 ) + (E − W )(E − W + 2m e c 2 ) − Q(Q + 2m e c 2 )
√
. (3.111)
2 E(E + 2m e c 2 ) (E − W )(E − W + 2m e c 2 )
The azimuthal scattering angle φ is sampled uniformly in the interval (0, 2π).
Hard close collisions of electrons
For the formulation of the sampling algorithm, it is convenient to introduce the reduced
energy loss κ ≡ W/E. The PDF of κ in close collisions of electrons with the k-th
oscillator is given by [see eqs. (3.69) and (3.70)]
( ) [
P (−)
1 1
k (κ) ≡ κ −2 F (−) (E, W ) Θ(κ − κ c ) Θ
2 − κ =
κ + 1
2 (1 − κ) 2
(
)]
( )
1
−
κ(1 − κ) + a 1
1
1 +
Θ(κ − κ c ) Θ
κ(1 − κ)
2 − κ , (3.112)