PENELOPE 2003 - OECD Nuclear Energy Agency

90 Chapter 3. Electron and positron interactions
( ∑ f k = Z) and the relation (3.52), we obtain
(
Ω 2 )
p
δ F ≃ ln
− 1, when β → 1. (3.64)
(1 − β 2 )I 2
The DCS for close collisions is given by
d 2 (
σ clo
dW dQ = 2πe4 ∑ 1 2m e c 2
f
m e v 2 k
W W (W + 2m e c 2 ) + β2 sin 2 )
θ clo
δ(W − Q) Θ(W − W
2m e c 2 k ),
k
where θ clo is the recoil angle, defined by eq. (3.40) with Q = W ,
We have
d 2 σ clo
dW dQ = 2πe4
m e v 2 ∑
k
cos 2 θ clo = W E
E + 2m e c 2
W + 2m e c 2 . (3.65)
(
)
1
f k 1 + β2 (E − W )W − EW
δ(W − Q) Θ(W − W
W 2 E(W + 2m e c 2 k ). (3.66)
)
DCS for close collisions of electrons
When the projectile is an electron, the DCS must be corrected to account for the indistinguishability
of the projectile and the target electrons. For distant interactions, the
effect of this correction is small (much smaller than the distortion introduced by our
modelling of the GOS) and will be neglected. The energy loss DCS for binary collisions
of electrons with free electrons at rest, obtained from the Born approximation with
proper account of exchange, is given by the Møller (1932) formula,
where
d 2 [ (
σ M
dW dQ = 2πe4 1 W
1 +
m e v 2 W 2 E − W
+a
) 2
− W
E − W
(
W
E − W + W 2
E 2 )]
δ(W − Q), (3.67)
(
)
E 2
( ) 2
γ − 1
a =
= . (3.68)
E + m e c 2 γ
To introduce exchange effects in the DCS for close interactions of electrons, we replace
the factor in parenthesis in eq. (3.66) by the analogous factor in Møller’s formula , i.e.
we take
d 2 σ (−)
clo
dW dQ = 2πe4 ∑ 1
f
m e v 2 k
W 2F (−) (E, W )δ(W − Q) Θ(W − W k ), (3.69)
with
k
( ) W 2
F (−) (E, W ) ≡ 1 +
− W
(
W
E − W E − W + a E − W + W 2 )
. (3.70)
E 2