PENELOPE 2003 - OECD Nuclear Energy Agency

B
'
B
'
3.1. Elastic collisions 79
1. 0
1. 0
A l (Z= 1 3 )
A u (Z= 7 9 )
0 . 8
el ec t r o n s
p o s i t r o n s
0 . 8
el ec t r o n s
p o s i t r o n s
0 . 6
A×(E/ 2 5 eV)
0 . 6
➤
A×(E/ 1 0 0 eV)
0 . 4
➤
0 . 4
➤
➤
0 . 2
0 . 2
0 . 0
0 . 0
- 0 . 2
1E+21E+3 1E+4 1E+5 1E+6 1E+7 1E+8 1E+9
E (eV)
- 0 . 2
1E+21E+3 1E+4 1E+5 1E+6 1E+7 1E+8 1E+9
E (eV)
Figure 3.5: Parameters of the MW model for scattering of electrons and positrons by aluminium
and gold atoms. The scale of the energy axes is logarithmic.
appropriate values of the total cross section and the first and second transport cross
sections. In penelope, these are calculated from atomic total and transport cross sections
by means of the additivity approximation (incoherent sum of scattered intensities).
This amounts to neglecting chemical binding effects. A more accurate approach, which
yields a good estimate of these effects, is provided by the following independent-atom
approximation (Walker, 1968; Yates, 1968). Assume that the interaction of the projectile
with each atom is still given by the free-atom static potential (3.4). The molecular
DCS may then be evaluated by adding the waves (not the currents) scattered from the
various atoms in the molecule and averaging over molecular orientations. The resulting
DCS is given by
dσ el
dΩ = ∑ i,j
sin(qa ij /¯h)
qa ij /¯h
[
fi (θ)f ∗ j (θ) + g i (θ)g ∗ j (θ) ] , (3.30)
where q = 2¯hk sin(θ/2) is the momentum transfer, a ij is the distance between the
atoms i and j and f i , g i are the scattering amplitudes, eq. (3.6), for the atom i. It
has been claimed that DCSs obtained from this formulation agree with experiments to
within ∼2% (Walker, 1968; Yates, 1968). DCSs for scattering of 100 eV and 2.5 keV
electrons in water vapour, obtained from the simple additivity rule and computed from
eq. (3.30), are compared in fig. 3.6. It is seen that, for energies above a few keV, chemical
binding causes a slight distortion of the DCS at small angles, and a slight rippling for
intermediate angles. Therefore, the use of the additivity approximation (i.e. neglecting