PENELOPE 2003 - OECD Nuclear Energy Agency

3.2. Inelastic collisions 85
plays a central role in the Bethe stopping power formula [eq. (3.103) below]. This
quantity has been determined empirically for a large number of materials (see Berger
and Seltzer, 1982, and references therein) from measurements of the stopping power of
heavy charged particles and/or from experimental optical dielectric functions. In the
following, we shall assume that the mean excitation energy of the stopping medium is
known.
3.2.1 GOS model
The simulation of inelastic collisions of electrons and positrons in penelope is performed
on the basis of the following GOS model, which is tailored to allow fast random
sampling of W and Q. We assume that the GOS splits into contributions from the different
atomic electron shells. Each atomic shell k is characterized by the number Z k of
electrons in the shell and the ionization energy U k . To model the contribution of a shell
to the GOS, we refer to the example of the hydrogen atom (fig. 3.7) and observe that for
Q > U k the GOS reduces to the Bethe ridge, whereas for Q < U k it is nearly constant
with Q and decreases rapidly with W ; a large fraction of the OOS concentrates in a
relatively narrow W -interval. Consideration of other well-known systems, such as inner
shells of heavy atoms (Manson, 1972) and the free-electron gas (Lindhard and Winther,
1964), shows that these gross features of the GOS are universal. Liljequist (1983) proposed
modelling the GOS of each atomic electron shell as a single “δ-oscillator”, which
is an entity with a simple GOS given by (see fig. 3.8)
F (W k ; Q, W ) = δ(W − W k )Θ(W k − Q) + δ(W − Q)Θ(Q − W k ), (3.47)
where δ(x) is the Dirac delta function and Θ(x) is the step function. The first term
represents resonant low-Q (distant) interactions, which are described as a single resonance
at the energy W k . The second term corresponds to large-Q (close) interactions,
in which the target electrons react as if they were free and at rest (W = Q). Notice
that the oscillator GOS satisfies the sum rule
∫ ∞
F (W k ; Q, W ) dW = 1 for any Q (3.48)
0
and, consequently, a δ-oscillator corresponds to one electron in the target. The Liljequist
GOS model for the whole atom is given by
df(Q, W )
dW
= ∑ k
f k [δ(W − W k )Θ(W k − Q) + δ(W − Q)Θ(Q − W k )] . (3.49)
where the summation in k extends over all bound electron shells (and the conduction
band, in the case of conductors) and the partial oscillator strength f k is identified with
the number of electrons in the k-th shell, i.e. f k = Z k . The corresponding OOS reduces
to
df(W )
dW
= ∑ f k δ(W − W k ), (3.50)
k