PENELOPE 2003 - OECD Nuclear Energy Agency

114 Chapter 3. Electron and positron interactions
Thus, the angular distribution in K reads
p(cos θ) = p d (cos θ ′ ) d(cos θ′ )
d(cos θ)
⎡
= A 3 8
⎣1 +
+ (1 − A) 3 4
( ) ⎤ 2
cos θ − β
⎦
1 − β cos θ
⎡
⎣1 −
( ) ⎤ 2
cos θ − β
⎦
1 − β cos θ
1 − β 2
(1 − β cos θ) 2
1 − β 2
(1 − β cos θ) 2 . (3.150)
Now, it is clear that when β tends to unity, the shape function concentrates at forward
directions.
We found that the benchmark partial-wave shape functions of Kissel et al. (1983)
can be closely approximated by the analytical form (3.150) if one considers A and β as
adjustable parameters. Explicitly, we write
p fit (cos θ) = A 3 8
⎡
⎣1 +
+ (1 − A) 3 4
( cos θ − β
′
⎡
1 − β ′ cos θ
⎣1 −
) 2
⎤
⎦
(
cos θ − β
′
1 − β ′ cos θ
1 − β ′2
(1 − β ′ cos θ) 2
) 2
⎤
⎦
1 − β ′2
(1 − β ′ cos θ) 2 , (3.151)
with β ′ = β(1 + B). The parameters A and B have been determined, by least squares
fitting, for the 144 combinations of atomic number, electron energy and reduced photon
energy corresponding to the benchmark shape functions tabulated by Kissel et al.
(1983). Results of this fit are compared with the original partial-wave shape functions
in fig. 3.16. The largest differences between the fits and the data were found for the
higher atomic numbers, but even then the fits are very accurate, as shown in fig. 3.16.
The quantities ln(AZβ) and Bβ vary smoothly with Z, β and κ and can be obtained
by cubic spline interpolation of their values for the benchmark cases. This permits the
fast evaluation of the shape function for any combination of Z, β and κ. Moreover, the
random sampling of the photon direction, i.e. of cos θ, can be performed by means of a
simple, fast analytical algorithm (see below). For electrons with kinetic energies larger
than 500 keV, the shape function is approximated by the classical dipole distribution,
i.e. by the analytical form (3.151) with A = 1 and β ′ = β.
3.3.4 Simulation of hard radiative events
Let us now consider the simulation of hard radiative events (W > W cr ) from the DCS
defined by eqs. (3.146) and (3.151). penelope reads the scaled bremsstrahlung DCS
from the database files and, by natural cubic spline interpolation/extrapolation in ln E,
produces a table for a denser logarithmic grid of 200 energies (and for the “standard”
mesh of 32 κ’s), which is stored in memory. This energy grid spans the full energy range