#
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| Geant 3.15 | GEANT User's Guide | PHYS330 ##
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Author(s) : L.Urban Submitted: 26.10.84
Origin : Same Revised: 17.12.92
+--------------------------------+
|CALL GDRELA |
+--------------------------------+
GDRELA initializes the ionization energy loss tables for different materials for muons, electrons and positrons and other particles. The energy binning is set within the array ELOW (/GCMULO/) in the routine GPHYSI. The tables are filled with the quantity dE/dx in GeV/cm (formula 4 below) which is valid for an element as well as a mixture or a compound. In the tables the dE/dx due to ionization is summed with the energy loss coming from Bremsstrahlung. For energy loss of electrons and positrons, it calls GDRELE using the following pointers:
JMA = LQ(JMATE-I) pointer to the I'th material JEL1 = LQ(JMA-1) pointer for dE/dx for electrons JEL1+90 pointer for dE/dx for positron
GDRELA is called at initialization time by GPHYSI.
+------------------------------------------------------+
| CALL GDRELE (EEL,CHARGE,JMA,DEDX) |
+------------------------------------------------------+
GDRELE computes the ionization energy loss (DEDX) for electrons (CHARGE = -1) and positrons (CHARGE = +1) with kinetic energy EEL in the material indicated by I in JMA = LQ(JMATE-I). It is called by the routine GDRELA.
+--------------------------------+
|CALL GDRSGA |
+--------------------------------+
CDRSGA calculates the total cross-section in all materials for delta rays - - - + for Moeller (e e ) and Bhabha (e e ) scattering and for muons. It tabulates the mean free path, lambda= ((1)/(Sigma)) (in cm) as a function of the medium and the energy. The energy binning is set within the array ELOW (common block /GCMULO/) in the routine GPHYSI. The following pointers are used (see JMATE data structure):
JMA = LQ(JMATE-I) pointer to the I'th material
JDRAY = LQ(JMA-11) pointer to Delta ray cross-sections
JDRAY pointer for electrons
JDRAY+90 pointer for positrons
+ -
JDRAY+180 pointer for mu /mu .
The routine is called at the initialization time by GPHYSI.
Let:
((dsigma(E, T))/(dT))
be the differential cross-section for the ejection of an electron with # kinetic energy T by an incident e of total energy E moving in a medium of density rho.
The variable DCUTE in common block /GCCUTS/ is the kinetic energy cut-off. Below this threshold the soft electrons ejected are # simulated as continuous energy loss by the incident e , and above it they are explicitly generated.
# The mean value of the energy lost by the incident e due to the soft electrons is:
DCUTE
E (E, DCUTE)= int ((dsigma(E,T))/(dT))T dT (1)
soft 0
whereas the total cross-section for the ejection of an electron of energy T>DCUTE is:
TMAX
sigma(E, DCUTE)= DCUTE((dsigma(E,T))/(dT)) dT (2)
where TMAX is the maximum energy transferable to the free electron:
+
TMAX ={ E- m for e . (3)
-
((E- m)/(2)) for e ,
where m is the electron mass. In this chapter, the method of calculation of the continuous energy loss and the total cross-section are explained. The next chapter (PHYS 331) deals with the explicit simulation of the delta rays.
The integration of (1) leads to the Berger-Seltzer formulae [bib-BERG], [bib-BETH], [bib-BLOC], [bib-EGS3], [bib-STER], [bib-MES1]:
2 2 2 #
((dE)/ (dx))= ((2pir mn)/(beta ))[ln((2(tau+2))/ ((I/m) ))+F (tau,Delta)- delta],(4)
0
where
gamma = ((E)/(m))
2 2
beta = 1-1/gamma
tau = gamma-1
tau = ((DCUTE)/(m))
c
#
DCUTE = energy cut for e
tau = maximum possible energy transfer
max + -
= tau for e , ((tau)/(2)) for e
Delta = min(tau ,tau )
c max
n = electron density of the medium
I = average mean ionisation energy
delta = density effect correction.
# The functions F are given by
+ 2 2 3 2 2 3 4 3
F (tau, Delta) = ln(tauDelta)-((Delta )/(tau))[tau +2Delta- ((3Delta y)/(2))-(Delta- ((Delta )/(3)))y - (((Delta )/(2))-tau((Delta )/ (3))+ ((Delta )/(4)))(5)
- 2 2 2
F (tau, Delta) = -1-beta +ln[(tau- Delta)Delta]+ tau/(tau-Delta) +[Delta / 2+(2tau +1) ln(1-Delta/tau)]1/ gamma , (6)
where y= 1/(gamma+ 1). The density effect correction is calculated as in (4):
0 if x
2
where x= ln(gamma -1)/2 ln10 The quantities n, I and the parameters of
the density effect correction (x ,x ,C, a,m) are computed in the routine
0 1
GPROBI, but we give the
corresponding formulae here. The electron density of the medium, n, can be
written as n= ((NrhoZ)/(A)) for elements and
n =((Nrho sum p Z )/ (sum p A ))for compounds/mixtures, [7]
i i i i i i
where
- N
- Avogadro's number
- Z(Z )
- atomic number (of i'th component) of the
i
medium
- A(A )
- atomic weight (of i'th component) of the
i
medium
- rho
- density of the material
- p
- proportion by number of the i'th element in the
i
material (for a mixture p can be calculated as p ' w /A where w the
i i i i i
corresponding proportion by weight).
The average mean ionization energy can be calculated as
0.9 -9
I =(16 #Z )10 Gev [8]
for a chemical element. In the case of a compound or mixture the average
value I= exp[((sum p Zln I(Z ))/(sum p Z ))] is used (1,2.3). The
i i i i i
density effect correction parameters can be computed (for condensed
medium, 4) as
# 9
C = C =1 +2 ln((I)/(28.8sqrt(rho)((sum p Z )/(sum p A ))10 ))
i i i i
m = 3
X = ((C)/ (2ln10))
a
m
a = ((2(ln 10)(X -X ))/ ((X -X )))
a 0 1 0
+---------+----------------------------------
| | | | |
| I | C | X | X |
+---------+-----------+-------0-------+--1--+
| | | | |
| -7 | | | |
| <10-7 | <3.681 | 0.2 | 2 |
| <10 | > =3.681 | - 0.326C-1 | 2 |
+---------+-----------+---------------+-----+
| -7 | | | |
| >=10 | 5. 215 | 0.2 | 3 |
| -7 | | | |
| >=10 | > =5.215 | -0. 326C-1.5 | 3 |
+---------+-----------+---------------+-----+
| |
Total cross-sections
The integration of formula [2] gives the total cross-section (3,5):
2 2 2 2
sigma(Z, E,DCUTE)= ((2pir mZ)/(beta (E-m)))[(((gamma-1))/ (gamma ))(((1)/(x))-1)+((1)/ (x))-((1)/(1- x))-((2gamma-1)/ (gamma ))ln((1-x)/ (x))](8)
0
- -
for Moeller (e e ) and
2 2 2 3
sigma(Z, E,DCUTE)= ((2pir mZ)/((E- m)))[((1)/(beta ))(((1)/ (x))-1)+B x+B (1-x)((beta )/(2))(1-x )+((beta )/ (3))(1-x )](9)
0 i 2 3 4
+ -
for Bhabha scattering (e e ), where
2 2
gamma =((E)/ (m)) beta = 1-((1)/(gamma ))
x =((DCUTE)/ (E-m)) gamma = ((1)/(gamma+ 1))
2 2
B =2- y B = (1-2y)(3+ y )
1 2
2 3 3
B =(1- 2y) + (1-2y) B = (1-2y)
3 4
The formulae [8] and [9] give the total cross-section of the scattering
above the threshold energies
thr thr
T =2DCUTE and T =DCUTE (10)
Moller Bhabha
The macroscopic cross-section is: sum = ((Nrhosigma(Z,E,DCUTE))/ (A)).
For a compound or mixture one has:
sum =((Nrho sum p sigma(Z , E,DCUTE))/(sum p A ))= Nrhosum (w A )* rho(Z ,E,DCUTE)
i i i i i i i i i i
- N
- Avogadro's number
- Z(Z )
- atomic number of the material (i th component
i
of the material)
- A(A )
- atomic weight of the material (i th
i
component)
- rho
- density of the material
- sigma
- total cross-section per atom for discrete
delta ray
- p
- proportion by number of the i th element in the
i
material p ' w /A where w is the corresponding proportion by weight
i i i i
- DCUTE
- energy cut off (DCUTE in the program).
The mean free path, lambda= ((1)/(Sigma)) (in cm) is tabulated at
initialization time as a function of the medium and of the energy by
routine GDRSGA.