CERN Program Library Long Writeup W5013 - CERNLIB ...

1. sample x from
1 1
ln 1 x c
x
setting
x = e r 1 ln x c
2. calculate the rejection function g(x) and:
• if r 2 >g(x) reject x and go back to 1;
• if r 2 ≤ g(x) accept x.
To apply the Migdal correction [76] all it has to be done is to multiply the rejection function by the Migdal
correction factor:
where
C M (ɛ) = 1+C 0/ɛ 2 c
1+C 0 /ɛ 2
C 0 = nr 0λ −2
, ɛ c = k c
π
E
n
r 0
λ−
electron density in the medium
classical electron radius
reduced Compton wavelength of the electron.
This correction decreases the cross-section for low photon energy.
After the successful sampling of ɛ, GBREME generates the polar angles of the radiated photon with respect
to the parent electron’s momentum. It is difficult to find in the literature simple formulas for this angle. For
example the double differential cross section reported by Tsai [13, 14] is the following:
{[ ]
dσ
dkdΩ = 2α2 e 2 2ɛ − 2
πkm 4 (1 + u 2 ) 2 + 12u2 (1 − ɛ)
(1 + u 2 ) 4 Z(Z +1)
[ ]
2 − 2ɛ − ɛ
2
+
(1 + u 2 ) 2 − 4u2 (1 − ɛ) [ ] }
(1 + u 2 ) 4 X − 2Z 2 f c ((αZ) 2 )
G el,in
u = Eθ
m
∫ m 2 (1+u 2 ) 2 [
] t −
X =
G el
Z(t)+G in tmin
Z (t)
t min
t 2 dt
Z
(t) atomic form factors
[
km 2 (1 + u 2 ] 2
)
t min =
=
2E(E − k)
[
ɛm 2 (1 + u 2 ] 2
)
2E(1 − ɛ)
This distribution is complicated to sample, and it is anyway only an approximation to within few percent,
if nothing else, due to the presence of the atomic form-factors. The angular dependence is contained in the
variable u = Eθm −1 . For a given value of u the dependence of the shape of the function on Z, E, ɛ = k/E
is very weak. Thus, the distribution can be approximated by a function
(
f(u) =C ue −au + due −3au) (2)
where
C =
9a2
9+d
a =0.625 d =0.13
(
0.8+ 1.3
Z
)(
100 + 1 )
(1 + ɛ)
E
285 PHYS341 – 3