PENELOPE 2003 - OECD Nuclear Energy Agency

3.3. Bremsstrahlung emission 117
This sampling algorithm is exact and very fast [notice that the binary search in step
(i) requires at most 5 comparisons], but is only applicable for the energies in the grid
where χ is tabulated.
To simulate bremsstrahlung emission by electrons with energies E not included in the
grid, we should first obtain the PDF p(E, κ) by interpolation along the energy axis and
then perform the random sampling of κ from this PDF using the algorithm described
above. This procedure is too time consuming. A faster method consists of assuming
that the grid of energies is dense enough so that linear interpolation in ln E is sufficiently
accurate. If E i < E < E i+1 , we can express the interpolated PDF as
with
p int (E, κ) = π i p(E i , κ) + π i+1 p(E i+1 , κ) (3.157)
π i = ln E i+1 − ln E
ln E i+1 − ln E i
, π i+1 = ln E − ln E i
ln E i+1 − ln E i
. (3.158)
These “interpolation weights” are positive and add to unity, i.e. they can be interpreted
as point probabilities. Therefore, to perform the random sampling of κ from p int (E, κ)
we can employ the composition method (section 1.2.5), which leads to the following
algorithm:
(i) Sample the integer variable I, which can take the values i or i + 1 with point
probabilities π i and π i+1 , respectively.
(ii) Sample κ from the distribution p int (E I , κ).
With this “interpolation by weight” method we only need to sample κ from the tabulated
PDFs, i.e. for the energies E i of the grid.
Angular distribution of emitted photons
The random sampling of cos θ is simplified by noting that the PDF given by eq. (3.151)
results from a Lorentz transformation, with speed β ′ , of the PDF (3.148). This means
that we can sample the photon direction cos θ ′ in the reference frame K ′ from the PDF
(3.148) and then apply the transformation (3.149) (with β ′ instead of β) to get the
direction cos θ in the laboratory frame.
To generate random values of cos θ from (3.151) we use the following algorithm,
which combines the composition and rejection methods,
(i) Sample a random number ξ 1 .
(ii) If ξ 1 < A, then
1) Sample a random number ξ and set cos θ ′ = −1 + 2ξ.
2) Sample a random number ξ.
3) If 2ξ > 1 + cos 2 θ ′ , go to 1).