CERN Program Library Long Writeup W5013 - CERNLIB ...

CALL GDRSGA
CDRSGA calculates the total cross-section for the production of δ-rays by electrons (Möller scattering),
positrons (Bhabha scattering) and muons. For hadrons, this value is calculated at tracking time. The mean
free path as a function of the energy is stored for every medium. The following pointers are used:
JMA = LQ(JMATE-I)
JDRAY = LQ(JMA-11)
JDRAY+NEK1
JDRAY+2*NEK1
pointer to the I th material;
δ-ray cross-section for electrons;
δ-ray cross-section for positrons;
δ-ray cross-section for muons.
The routine is called during initialisation by GPHYSI.
2 Method
In GEANT, δ-rays are generated only above a threshold energy T cut which corresponds to the variable DCUTE
for electrons and positrons, DCUTM for all other charged particles. The total cross-section for the production
of a δ-ray electron of kinetic energy T>T cut =DCUTE, DCUTM by a particle of kinetic energy E is:
σ(E,T cut )=
∫ Tmax
T cut
dσ(E,T)
dT (1)
dT
where T max is the maximum energy transferable to the free electron:
2m(γ 2 − 1)
T max =
1+2γ m M + ( m ) 2
(2)
M
The energy lost in ionising collisions producing δ-rays below T cut are included in the continuous energy
loss. The mean value of the energy lost due to these soft collisions is:
∫ Tcut
E loss (E,T cut )= T dσ(E,T) dT (3)
0 dT
where m is the electron mass and M is the mass of the incident particle.
In this chapter, the method of calculation of the continuous energy loss and the total cross-section are
explained. The generation of δ-rays is explained in chapter [PHYS331].
2.1 Continuous energy loss
The integration of (3) leads to the Bethe-Block stopping power or to the restricted energy loss formula [55]:
⎧ [ ( )
( ) D ZZ2 inc 2me β 2 γ 2 T max
1 dE
⎪⎨ Aβ 2 ln
I 2 − 2β 2 − δ − 2C ]
e
if T cut ≥ T max
Z
=
[ (
ρ dx
⎪⎩ D ZZ2 inc 2me β 2 γ 2 ) (
T c
Aβ 2 ln
I 2 − β 2 1+ T )
c
− δ − 2C ]
(4)
e
if T cut