PENELOPE 2003 - OECD Nuclear Energy Agency

96 Chapter 3. Electron and positron interactions
1Ε+10
1Ε+10
electrons
positrons
e l e c trons
1Ε+9
1Ε+9
S in
/ρ (eV cm 2 /g)
1Ε+8
Au (×100)
Ag (×10)
S in
/ρ (eV cm 2 /g)
1Ε+8
Au (×100)
Ag (×10)
1Ε+7
1Ε+7
Al
Al
1Ε+6
1Ε+6
1Ε+21Ε+3 1Ε+4 1Ε+5 1Ε+6 1Ε+7 1Ε+8 1Ε+9
E (eV)
1Ε+21Ε+3 1Ε+4 1Ε+5 1Ε+6 1Ε+7 1Ε+8 1Ε+9
E (eV)
Figure 3.10: Collision stopping power S in /ρ for electrons and positrons in aluminium, silver
(×10) and gold (×100) as a function of the kinetic energy. Continuous and dashed curves are
results from the present model. Crosses are data from the ICRU37 tables (1984) [also, Berger
and Seltzer, 1982)]. The dotted curves are predictions from the Bethe formula (3.103), for
electrons and positrons.
3.2.4 Stopping power of high-energy electrons and positrons
It is of interest to evaluate explicitly the stopping power for projectiles with high energies
(E ≫ U k ). We shall assume that U k ≪ 2m e c 2 (for the most unfavourable case of the
K shell of heavy elements, U k is of the order of 2m e c 2 /10). Under these circumstances,
Q − ≪ 2m e c 2 and we can use the approximation [see eq. (A.35)]
Q − ≃ W 2 k /(2m e c 2 β 2 ). (3.98)
The contribution from distant (longitudinal and transverse) interactions to the stopping
cross section is then [see eqs. (3.77) and (3.78)]
σ (1)
dis ≃ 2πe4 ∑
m e v 2
k
f k
{ln
The contribution of close interactions is given by
σ (1)
clo = 2πe4 ∑
m e v 2
k
( 2me c 2 ) ( )
}
1
+ ln − β 2 − δ
W k 1 − β 2 F . (3.99)
f k
∫ Wmax
W k
W −1 F (±) (E, W ) dW. (3.100)