PENELOPE 2003 - OECD Nuclear Energy Agency

3.1. Elastic collisions 77
becomes too “wide”. This suggests using a modified Wentzel (MW) model obtained by
combining a Wentzel distribution with a simple distribution, which takes different forms
in these two cases,
• Case I. If 〈µ 2 〉 W,A0 > 〈µ 2 〉 (the Wentzel distribution is too wide), we take p MW (µ) as a
statistical admixture of the Wentzel distribution and a delta distribution (a zero-width,
fixed scattering angle process)
with
p MW,I (µ) = (1 − B) p W,A (µ) + B δ(µ − 〈µ〉) (3.24)
A = A 0 and B = 〈µ2 〉 W,A − 〈µ 2 〉
〈µ 2 〉 W,A − 〈µ〉 2 . (3.25)
Notice that in this case we usually have 〈µ〉 ≪ 1, so that the delta distribution is at
very small angles. Although we have introduced a discrete peak in the DCS, its effect
is smeared out by the successive collisions and not visible in the multiple scattering
angular distributions.
• Case II. If 〈µ 2 〉 W,A0 < 〈µ 2 〉 (the Wentzel distribution is too narrow), we express p MW (µ)
as a statistical admixture of a Wentzel distribution (with A not necessarily equal to A 0 )
and a triangle distribution in the interval (1/2,1),
p MW,II (µ) = (1 − B) p W,A (µ) + B 8 (µ − 1/2) Θ (µ − 1/2) . (3.26)
The parameters A and B are obtained from the conditions (3.19) and (3.20), which give
From the first of these equations,
(1 − B) 〈µ〉 W,A + B 5 6 = 〈µ〉
(1 − B) 〈µ 2 〉 W,A + B 17
24 = 〈µ2 〉. (3.27)
B =
Inserting this value in the second of eqs. (3.27), we obtain
〈µ〉 − 〈µ〉 W,A
(5/6) − 〈µ〉 W,A
. (3.28)
( )
( )
17
5
24 − 〈µ2 〉 〈µ〉 W,A −
6 − 〈µ〉 〈µ 2 〉 W,A = 17
24 〈µ〉 − 5 6 〈µ2 〉. (3.29)
For all situations of interest, this equation has a single root A in the interval (0, A 0 ) and
can be easily solved by means of a bipartition procedure. The value of B given by eq.
(3.28) is then positive and less than unity, as required.
In fig. 3.4 we compare partial-wave DCSs and MW model DCSs for elastic scattering
of electrons of various energies by gold atoms. The considered energies correspond to
the case-II MW model [so that the distribution p MW (µ) is continuous]. We see that
the MW model does imitate the partial wave DCSs, but the differences are significant.