PENELOPE 2003 - OECD Nuclear Energy Agency

88 Chapter 3. Electron and positron interactions
(with W < W cc ) is described by means of a multiple scattering approximation, which
does not require detailed knowledge of the shell DCSs. Hard collisions may produce
ionizations in deep electron shells, which leave the target atom in a highly excited
state (with a vacancy in an inner shell) that decays by emission of energetic x-rays and
Auger electrons. In penelope we use the GOS model only to describe the effect of the
interactions on the projectile and the emission of knock-on secondary electrons. K and
L shells with ionization energy U j larger than max(200 eV, W cc ) will be referred to as
“inner” shells. The production of vacancies in these inner shells is simulated by means
of more accurate ionization cross sections (see section 3.2.6). Electron shells other than
K and L shells, or with U j < max(200 eV, W cc ), will be referred to as “outer” shells.
The present theory is directly applicable to compounds (and mixtures), since the
oscillators may pertain either to atoms or molecules. When the value of the mean
excitation energy of the compound is not known, it may be estimated from Bragg’s
additivity rule as follows. Consider a compound X x Y y , in which the molecules consist
of x atoms of the element X and y atoms of the element Y. The number of electrons
per molecule is Z M = xZ X + yZ Y , where Z X stands for the atomic number of element
X. According to the additivity rule, the GOS of the compound is approximated as the
sum of the atomic GOSs of the atoms so that
Z M ln I = xZ X ln I X + yZ Y ln I Y , (3.56)
where I X denotes the mean excitation energy of element X.
For heavy elements, and also for compounds and mixtures with several elements, the
number of electron shells may be fairly large (of the order of sixty for an alloy of two
heavy metals). In these cases, it would be impractical to treat all shells with the same
detail/accuracy. In fact, the description of the outer shells can be simplified without
sacrificing the reliability of the simulation results. In penelope, the maximum number
of oscillators for each material is limited. When the number of actual shells is too large,
oscillators with similar resonance energies are grouped together and replaced by a single
oscillator with oscillator strength equal to the sum of strengths of the original oscillators.
The resonance energy of the group oscillator is set by requiring that the contribution to
the mean excitation energy I equals the sum of contributions of the grouped oscillators;
this ensures that grouping will not alter the stopping power of fast particles (with E
substantially greater than the ionization energy of the grouped oscillators).
3.2.2 Differential cross sections
The DCS for inelastic collisions obtained from our GOS model can be split into contributions
from distant longitudinal, distant transverse and close interactions,
d 2 σ in
dW dQ = d2 σ dis,l
dW dQ + d2 σ dis,t
dW dQ + d2 σ clo
dW dQ . (3.57)