PENELOPE 2003 - OECD Nuclear Energy Agency

110 Chapter 3. Electron and positron interactions
The radiative DCS for positrons reduces to that of electrons in the high-energy limit
but is smaller for intermediate and low energies. Owing to the lack of more accurate
calculations, the DCS for positrons is obtained by multiplying the electron DCS by a
κ-independent factor, i.e.
dσ (+)
br
= F p(Z, E) dσ(−) br
dW . (3.138)
dW
The factor F p (Z, E) is set equal to the ratio of the radiative stopping powers for positrons
and electrons, which has been calculated by Kim et al. (1986) (cf. Berger and Seltzer,
1982). In the calculations we use the following analytical approximation
where
F p (Z, E) = 1 − exp(−1.2359 × 10 −1 t + 6.1274 × 10 −2 t 2 − 3.1516 × 10 −2 t 3
+ 7.7446 × 10 −3 t 4 − 1.0595 × 10 −3 t 5 + 7.0568 × 10 −5 t 6
− 1.8080 × 10 −6 t 7 ), (3.139)
t = ln
(
1 + 106
Z 2
)
E
. (3.140)
m e c 2
Expression (3.139) reproduces the values of F p (Z, E) tabulated by Kim et al. (1986) to
an accuracy of about 0.5%.
3.3.2 Integrated cross sections
The total cross section for bremsstrahlung emission is infinite due to the divergence of
the DCS (3.131) for small reduced photon energies. Nevertheless, the cross section for
emission of photons with reduced energy larger than a given cutoff value W cr is finite.
The corresponding mean free path is
∫ E
∫
λ −1
br (E; W dσ br Z2 1 1
cr) ≡ N dW = N χ(Z, E, κ) dκ, (3.141)
W cr dW β 2 κ cr κ
where κ cr = W cr /E. The radiative stopping power and the radiative energy straggling
parameter, defined by
and
S br (E) ≡ N
Ω 2 br (E) ≡ N ∫ E
0
∫ E
0
W 2 dσ br
dW
W dσ br
dW
∫
Z2 1
dW = N
β E χ(Z, E, κ) dκ (3.142)
2 0
∫
Z2 1
dW = N
β 2 E2 κ χ(Z, E, κ) dκ, (3.143)
0
are both finite. For the kinetic energies E i of the grid, these quantities are easily calculated
from the tabulated scaled DCS by using linear interpolation in κ. For positrons,
the definitions (3.141)-(3.143) must be multiplied by the factor F p (Z, E) [eq. (3.139)].
Radiative stopping powers of aluminium, silver and gold for electrons and positrons
are shown as functions of the kinetic energy in fig. 3.14. The stopping powers computed