PENELOPE 2003 - OECD Nuclear Energy Agency

2.4. Electron-positron pair production 59
where
v = (0.2840 − 0.1909a) ln(4/κ) + (0.1095 + 0.2206a) ln 2 (4/κ)
+ (0.02888 − 0.04269a) ln 3 (4/κ) + (0.002527 + 0.002623a) ln 4 (4/κ). (2.89)
Then, the single quantity η ∞ characterizes the triplet production for each element.
The approximation given by eq. (2.77) with the fitted value of the screening radius,
fails at low energies where it systematically underestimates the total cross section (it
can even become negative). To compensate for this fact we introduce an empirical
correcting term F 0 (κ, Z), which acts in a way similar to the Coulomb correction. To
facilitate the random sampling, the Bethe-Heitler DCS, eq. (2.77), including this lowenergy
correction and a high-energy radiative correction, is written in the form
[ ( )
dσ pp
= r 2 2 1 2
]
dɛ
eαZ[Z + η] C r 2
3 2 − ɛ φ 1 (ɛ) + φ 2 (ɛ) , (2.90)
where
with
φ 1 (ɛ) = g 1 (b) + g 0 (κ)
φ 2 (ɛ) = g 2 (b) + g 0 (κ) (2.91)
g 1 (b) = 1 2 (3Φ 1 − Φ 2 ) − 4 ln(Rm e c/¯h) = 7 3 − 2 ln(1 + b2 ) − 6b arctan(b −1 )
− b 2 [ 4 − 4b arctan(b −1 ) − 3 ln(1 + b −2 ) ] ,
g 2 (b) = 1 4 (3Φ 1 + Φ 2 ) − 4 ln(Rm e c/¯h) = 11 6 − 2 ln(1 + b2 ) − 3b arctan(b −1 )
+ 1 2 b2 [ 4 − 4b arctan(b −1 ) − 3 ln(1 + b −2 ) ] ,
g 0 (κ) = 4 ln(Rm e c/¯h) − 4f C (Z) + F 0 (κ, Z). (2.92)
C r = 1.0093 is the high-energy limit of Mork and Olsen’s radiative correction (Hubbell
et al., 1980).
The correcting factor F 0 (κ, Z) has been determined by requiring that the total cross
section for pair production obtained from the expression given in eq. (2.90) (with η = 0)
coincides with the total cross sections for pair production in the field of the nucleus tabulated
by Hubbell et al. (1980). By inspection and numerical fitting, we have obtained
the following analytical approximation
F 0 (κ, Z) = (−0.1774 − 12.10a + 11.18a 2 )(2/κ) 1/2
+ (8.523 + 73.26a − 44.41a 2 )(2/κ)
− (13.52 + 121.1a − 96.41a 2 )(2/κ) 3/2
+ (8.946 + 62.05a − 63.41a 2 )(2/κ) 2 . (2.93)