PENELOPE 2003 - OECD Nuclear Energy Agency

A
l
A
u
3.2. Inelastic collisions 97
1Ε+3
ρλ in
(µg /cm 2 ), S in
/ρ (eV cm 2 /µg)
1Ε+2
1Ε+1
1Ε+0
S in
/ρ
ρλ in
ρλ in
(µg /cm 2 ), S in
/ρ (eV cm 2 /µg)
1Ε+2
1Ε+1
1Ε+0
S in
/ρ
ρλ in
W cb
= 30 eV
W cb
= 4
0 eV
1Ε− 1
1Ε+2 1Ε+3 1Ε+4 1Ε+5
E (eV)
1Ε+2 1Ε+3 1Ε+4 1Ε+5
E (eV)
Figure 3.11: Collision mean free path and stopping power for low-energy electrons in aluminium
and gold. The plotted quantities are ρλ in and S in /ρ. Special symbols are experimental
data from different sources (see Fernández-Varea et al., 1993a); closed symbols for mean free
paths and open symbols for stopping powers.
Recalling that E ≫ U k , we have
σ (1)
clo
≃ 2πe4 ∑
m e v 2
+ 1 8
k
( ) [
]
E
f k
{ln + 1 − 1 + β 2 + 2
√1 − β
W 2 ln 2
k
( √ ) } 2
1 − 1 − β 2
(3.101)
for electrons and
σ (1)
clo ≃ 2πe4
m e v 2 ∑
k
( ) E
f k
{ln − b 1 + b 2
W k 2 − b 3
3 + b }
4
4
(3.102)
for positrons. Adding the distant and close stopping cross sections, and using the relation
(3.52), we arrive at the familiar Bethe formula for the stopping power,
S in ≡ N ( σ (1)
dis + ) 2πe 4 { ( ) }
E
2
γ + 1
σ(1) clo = N
m e v 2Z ln
+ f (±) (γ) − δ
I 2 F , (3.103)
2
where
f (−) (γ) = 1 − β 2 − 2γ − 1
γ 2 ln 2 + 1 8
( ) 2
γ − 1
(3.104)
γ