PENELOPE 2003 - OECD Nuclear Energy Agency

54 Chapter 2. Photon interactions
(ii) Sample τ from P i (τ) using the sampling formulae
⎧
⎪⎨ τ ξ min if i = 1,
τ =
⎪⎩ [τmin 2 + ξ (1 − τmin)] 2 1/2 if i = 2,
(2.70)
which can be easily derived by the inverse transform method (section 1.2.2).
(iii) Determine cos θ using eq. (2.62),
and compute the quantities p i,max (E, θ), eq. (2.49), and
cos θ = 1 − 1 − τ
κτ , (2.71)
S(E, θ) = ∑ i
f i Θ(E − U i ) n A i (p i,max ). (2.72)
(iv) Generate a new random number ξ.
(v) If ξ > T (cos θ), go to step (i).
(vi) Deliver cos θ.
The efficiency of this algorithm, i.e. the probability of accepting a generated cos θ-value,
increases monotonically with photon energy and is nearly independent of Z; typical
values are 35%, 80% and 95% for E = 1 keV, 1 MeV and 10 MeV, respectively.
Once the direction of the emerging photon has been set, the active electron shell i
is selected with relative probability equal to Z i Θ(E − U i ) n A i (p i,max (E, θ)). A random
value of p z is generated from the analytical Compton profile (2.57) using the sampling
formula (2.59). If p z is less than −m e c, it is rejected and a new shell and a p z -value
are sampled 3 . Finally, the factor F (p z ) in the PDF (2.46) is accounted for by means of
a rejection procedure. It should be noted that the approximation F ≃ 1 is valid only
when the DCS is integrated over E ′ ; otherwise the complete expression (2.44) must be
used. Let F max denote the maximum value of F (p z ), which occurs at p z = 0.2m e c or
−0.2m e c; a random number ξ is generated and the value p z is accepted if ξF max < F (p z ),
otherwise the process of selecting a shell and a p z -value is reinitiated. The energy E ′ of
the emerging photon is then calculated from eq. (2.36), which gives
E ′ = E
where
[
√
]
τ
(1 − tτ cos θ) + sign(p
1 − tτ 2 z ) (1 − tτ cos θ) 2 − (1 − tτ 2 )(1 − t) , (2.73)
t ≡ (p z /m e c) 2 and sign(p z ) ≡ p z /|p z |. (2.74)
For photons with energy larger than 5 MeV, for which Doppler broadening is negligible,
we set E ′ = E C (which amounts to assuming that p z = 0). In this case, the active
3 Notice that, due to the approximation introduced in eq. (2.36), a value p z < −m e c would yield a
negative energy for the scattered photon.