PENELOPE 2003 - OECD Nuclear Energy Agency

82 Chapter 3. Electron and positron interactions
on the basis of the first-order (plane-wave) Born approximation. The extension of the
theory to inelastic collisions in condensed materials has been discussed by Fano (1963).
The formal aspects of the quantum theory for condensed matter are quite complicated.
Fortunately, the results are essentially equivalent to those from classical dielectric theory.
The effect of individual inelastic collisions on the projectile is completely specified
by giving the energy loss W and the polar and azimuthal scattering angles θ and φ,
respectively. For amorphous media with randomly oriented atoms (or molecules), the
DCS for inelastic collisions is independent of the azimuthal scattering angle φ. Instead
of the polar scattering angle θ, it is convenient to use the recoil energy Q [see eqs. (A.29)
and (A.30)], defined by
Q(Q + 2m e c 2 ) = (cq) 2 . (3.38)
The quantity q is the magnitude of the momentum transfer q ≡ p − p ′ , where p and p ′
are the linear momenta of the projectile before and after the collision. Notice that Q is
the kinetic energy of an electron that moves with a linear momentum equal to q.
Let us first consider the inelastic interactions of electrons or positrons (z0 2 = 1) with
an isolated atom (or molecule) containing Z electrons in its ground state. The DCS
for collisions with energy loss W and recoil energy Q, obtained from the first Born
approximation, can be written in the form (Fano, 1963)
d 2 σ in
dW dQ = 2πz2 0e 4 (
2m e c 2
m e v 2 W Q(Q + 2m e c 2 ) + β2 sin 2 θ r W 2m e c 2 )
df(Q, W )
[Q(Q + 2m e c 2 ) − W 2 ] 2 dW , (3.39)
where v = βc is the velocity of the projectile. θ r is the angle between the initial
momentum of the projectile and the momentum transfer, which is given by eq. (A.42),
cos 2 θ r =
W 2 /β 2 (
1 + Q(Q + 2m ec 2 ) − W 2 ) 2
. (3.40)
Q(Q + 2m e c 2 ) 2W (E + m e c 2 )
The result (3.39) is obtained in the Coulomb gauge (Fano, 1963); the two terms on the
right-hand side are the contributions from interactions through the instantaneous (longitudinal)
Coulomb field and through the exchange of virtual photons (transverse field),
respectively. The factor df(Q, W )/dW is the atomic generalized oscillator strength
(GOS), which completely determines the effect of inelastic interactions on the projectile,
within the Born approximation. Notice, however, that knowledge of the GOS does not
suffice to describe the energy spectrum and angular distribution of secondary knock-on
electrons (delta rays).
The GOS can be represented as a surface over the (Q, W ) plane, which is called the
Bethe surface (see Inokuti, 1971; Inokuti et al., 1978). Unfortunately, the GOS is known
in analytical form only for two simple systems, namely, the (non-relativistic) hydrogenic
ions (see fig. 3.7) and the free-electron gas. Even in these cases, the analytical expressions
of the GOSs are too complicated for simulation purposes. For ionization of inner shells,
the GOS can be computed numerically from first principles (see e.g. Manson, 1972), but
using GOSs defined through extensive numerical tables is impractical for Monte Carlo
simulation. Fortunately, the physics of inelastic collisions is largely determined by a few