PENELOPE 2003 - OECD Nuclear Energy Agency

86 Chapter 3. Electron and positron interactions
which has the same form (a superposition of resonances) as the OOS used by Sternheimer
(1952) in his calculations of the density effect correction. In order to reproduce the
high-energy stopping power given by the Bethe formula (Berger and Seltzer, 1982), the
oscillator strengths must satisfy the Bethe sum rule (3.45),
∑
k f k = Z, (3.51)
and the excitation energies must be defined in such a way that the GOS model leads,
through eq. (3.46), to the accepted value of the mean excitation energy I,
∑
k f k ln W k = Z ln I. (3.52)
As the partial oscillator strength f k has been set equal to the number of electrons in the
k-th shell, the Bethe sum rule is automatically satisfied.
W
W m
(Q)
W=Q
W k
E/ 2
U k
Q
−
Q
Figure 3.8: Oscillator model for the GOS of an inner shell with U k = 2 keV. The continuous
curve represents the maximum allowed energy loss as a function of the recoil energy, W m (Q),
for electrons/positrons with E = 10 keV. For distant interactions the possible recoil energies lie
in the interval from Q − to W k . Recoil energies larger than W k correspond to close interactions.
The largest allowed energy loss W max is E/2 for electrons and E for positrons (see text).
The largest contribution to the total cross section arises from low-W (soft) excitations.
Therefore, the total cross section is mostly determined by the OOS of weakly
bound electrons, which is strongly dependent on the state of aggregation. In the case
of conductors and semiconductors, electrons in the outermost shells form the conduction
band (cb). These electrons can move quite freely through the medium and, hence,
their binding energy is set to zero, U cb = 0. Excitations of the conduction band will be
described by a single oscillator, with oscillator strength f cb and resonance energy W cb .
These parameters should be identified with the effective number of electrons (per atom
or molecule) that participate in plasmon excitations and the plasmon energy, respectively.
They can be estimated e.g. from electron energy-loss spectra or from measured