CERN Program Library Long Writeup W5013 - CERNLIB ...

Geant 3.16 GEANT User’s Guide PHYS328
Origin : G.Lynch Submitted: 23.08.93
Revision : Revised: 16.12.93
Documentation : F.Carminati
1 Subroutines
Plural scattering
CALL GMCOUL
(OMEGA,DIN*)
OMEGA
DIN
(REAL) Ω 0 of the Molière theory;
(REAL) array of dimension 3 containing the direction of the scattered particle with respect to
the incoming direction;
Generates single scatters in small angle approximation returning the new direction with respect to the old
one. It is called by GMULTS when Ω 0 ≤ 20.
2 Method
In the Molière theory the average number of Coulomb scatters for a charged particle in a step is expressed
by the parameter Ω 0 (see [PHYS325]). When Ω 0 ≤ 20, the Molière theory is not applicable any more, even
if it has been noted [48] that it still gives reasonable results down to its mathematical limit Ω 0 = e. The
range 1 < Ω 0 ≤ 20 is called the plural scattering regime. An interesting study of this regime can be found
in [49].
In GEANT, when Ω 0 ≤ 20, a direct simulation method is used for the scattering angle. The number of
scatters is distributed according to a Poissonian law with average Ω=kΩ 0 = e 2γ−1 Ω 0 ≈ 1.167Ω 0 with
γ =0.57721 ... Euler’s constant. Using the customary notations for the probability distribution function
for small angle (sin θ ≈ θ) single scattering, we can write:
f(θ)θdθ = dσ 1
θdθ (1)
θdθ σ
where σ is the cross section for single elastic scattering. We use as cross section the one reported by Molière
[43] [44]:
(
dσ
θdθ =2π 2ZZinc e 2 ) 2
1
pv (θ 2 + χ 2 α )2
This is the classical Rutherford cross section corrected by the screening angle χ α . This angle is described
by Molière as a correction to the Born approximation used to derive the quantum mechanical form of the
Rutherford cross section. We have then:
∫ ( ) ∞
dσ
σ =
0 θdθ θdθ =2π 2ZZinc e 2 2 ∫ ( ) ∞
θdθ
pv 0 (θ 2 + χ 2 α) 2 = π 2ZZinc e 2 2
1
pv χ 2 α
and so equation (1) becomes:
f(θ)θdθ = χ 2 2θdθ
α
(θ 2 + χ 2 α) 2 = 2ΘdΘ
(1 + Θ 2 ) 2
250 PHYS328 – 1