PENELOPE 2003 - OECD Nuclear Energy Agency

4.3. Combined scattering and energy loss 145
of the variable µ as
dσ dis,l
dµ = 4πe4
m e v 2 ∑
k
f k
1
W k
m e c 2
4 cp cp k µ + (cp − cp k ) 2 4 cp cp k
2(Q + m e c 2 ) . (4.103)
Considering that Q ≪ m e c 2 for the majority of soft distant collisions, we have
where
and
dσ dis,l
dµ = 2πe4
m e v 2 ∑
k
f k
1
W k
1
R k + µ , 0 < µ < µ k, (4.104)
R k = (cp − cp k) 2
4 cp cp k
(4.105)
µ k = µ(Q = W k ) = W k(W k + 2m e c 2 ) − (cp − cp k ) 2
4cp cp k
. (4.106)
On the other hand, the DCS per unit oscillator strength for soft (W < W cc ) close
collisions with the i-th oscillator is given by [see eqs. (3.69) and (3.75)]
dσ (±)
clo
dW
= 2πe4 ∑
m e v 2
k
f k
1
W 2 F (±) (E, W ). (4.107)
The angular deflection and the energy loss are related by (3.116), which implies that
and
Therefore,
W = E(E + 2m ec 2 )2(µ − µ 2 )
2E(µ − µ 2 ) + m e c 2 (4.108)
dW
dµ = E(E + 2m ec 2 )m e c 2 2(1 − 2µ)
[2E(µ − µ 2 ) + m e c 2 ] 2 . (4.109)
where
with
dσ (±)
clo
dµ = 2πe4
m e v 2 ∑
k
f k
1
W 2 F (±) (E, W ) dW
dµ , µ k < µ < µ cc , (4.110)
µ cc = µ(Q = W cc ) = W cc(W cc + 2m e c 2 ) − (cp − cp cc ) 2
4cp cp cc
(4.111)
(cp cc ) 2 = (E − W cc )(E − W cc + 2m e c 2 ). (4.112)
The angular DCS for soft inelastic interactions is then given by
dσ s
dµ = dσ dis,l
dµ + dσ(±) clo
dµ
{
= 2πe4 ∑ 1 1
f
m e v 2 k
W k R k + µ + 1
W F (±) (E, W ) dW
2 dµ
k
}
, (4.113)