PENELOPE 2003 - OECD Nuclear Energy Agency

3.2. Inelastic collisions 83
global features of the Bethe surface. Relatively simple GOS models can be devised that
are consistent with these features and, therefore, lead to a fairly realistic description of
inelastic interactions (see e.g. Salvat and Fernández-Varea, 1992).
.
2.0
1.5
1.0
0.5
d f ( Q,W ) / dW
1
2
3
4
5
W/U i
6
7
8
2
1
0
−1
ln (Q/U i )
0.0
−2
Figure 3.7: The GOS for ionization of the hydrogen atom (Z = 1) in the ground state.
All energies are in units of the ionization energy U i = 13.6 eV. The GOS for ionization of
(non-relativistic) hydrogenic ions is independent of Z if energies are expressed in units of the
ionization energy.
As mentioned above, the “atomic” DCS for inelastic interactions in dense media
can be obtained from a semiclassical treatment in which the medium is considered as
a dielectric, characterized by a complex dielectric function ɛ(k, ω), which depends on
the wave number k and the frequency ω. In the classical picture, the (external) electric
field of the projectile polarizes the medium producing an induced electric field that
causes the slowing down of the projectile. The dielectric function relates the Fourier
components of the total (external+induced) and the external electric potentials. It is
convenient to interpret the quantities q = ¯hk and W = ¯hω as the momentum and energy
transfers and consider that the dielectric function depends on the variables Q [defined
by eq. (3.38)] and W . The DCSs obtained from the dielectric and quantum treatments
are consistent (i.e. the former reduces to the latter for a low-density medium) if one
assumes the identity
df(Q, W )
dW ≡ W Q + m ec 2
m e c 2
2Z
πΩ 2 p
Im
( ) −1
, (3.41)
ɛ(Q, W )
where Ω p is the plasma energy of a free-electron gas with the electron density of the