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| Geant 3.16 | GEANT User's Guide | PHYS340 ##
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Author(s) : L. Urban Submitted: 28.05.86
Origin : Same Revised: 03.05.93
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|CALL GBRELA |
+--------------------------------+
GBRELA fills the tables for the energy loss of electrons, positrons and muons due to Bremsstrahlung at initialization time for different materials. The energy binning is set within the array ELOW (COMMON CGMULO) in the routine GPHYSI. In the tables, the dE/dx due to Bremsstrahlung is summed with that due to the ionization. For energy loss of electrons and positrons, GBRELA calls the function GBRELE. Following pointers are used:
GBRELA is called at initialization time by GPHYSI.
VALUE = GBRELE(ZZ,T,BCUT)
GBRELE calculates the energy loss due to Bremsstrahlung of an electron with kinetic energy T in material with atomic number ZZ. It is called by GBRELA and for energies below the cut BCUT it adds the contribution of Bremsstrahlung to. Above this cut, the Bremsstrahlung process is simulated explicitly (see PHYS 341) and tabulation of these continuous losses is not needed. GBRELE is called by GBRELA.
VALUE = GBFLOS(T,C)
GBFLOS calculates a weight
factor for the positron continuous bremsstrahlung energy loss. T is the
kinetic energy in GeV of the positron and C is the energy cut for
Bremsstrahlung (BCUTE). The output is the ratio of positron to electron
brems loss so that:
calculates the total cross-section for Bremsstrahlung in all materials. It
tabulates the mean free path, lambda= ((1)/(Sigma)) (in cm) as a function
of medium and energy. The energy binning is set within the array ELOW
(COMMON CGMULO) in the routine
GPHYSI. The following pointers
are used:
GBRSGE calculates the total cross-section of Bremsstrahlung of an electron
with kinetic energy T in material with atomic number ZZ. It is called by
GBRSGA. For kinetic energies which are below the cut BCUT or for which
Bremsstrahlung process is not simulated explicitly (see PHYS341) it
returns 0.
GBFSIG calculates a weight
factor for the positron discrete (hard) bremsstrahlung cross section. T is
the kinetic energy in GeV of the positron and C is the energy cut for
Bremsstrahlung (BCUTE). The output is the ratio of positron to electron
brems cross-section so that:
Let's call dsigma(Z,T,k)/dk the differential cross-section for production
of a photon of energy k by an electron of kinetic energy T in the field of
an atom of charge Z, and k the energy cut-off below which the soft
c
photons are treated as continuous energy lost (BCUTE in the program). Then
the mean value of the energy lost by the electron due to soft photons is
whereas the total cross-section for the emission of a photon of energy
>k is
c
Many theories of the Bremsstrahlung process exist, each with its own
limitations and regions of applicability. Perhaps the best synthesis of
these theories can be found in the paper of S.M. Seltzer and M.J. Berger
[bib-SEL1]. The authors give a
tabulation of the Bremsstrahlung cross-section dsigma/dk, differential in
the photon energy k, for electrons with kinetic energies T from 1 keV to
10 GeV. For electron energies above 10 GeV the screened Bethe-Heitler
differential cross-section can be used [bib-EGS3], [bib-NELS] together with the Midgal
corrections [bib-MES1], [bib-MIGD]. The first of the two
Migdal corrections is important for very high electron energies only (T>=
1 TeV) and has the effect of reducing the cross-section. The second Migdal
correction is effective even at ``ordinary'' energies (100 MeV -- 1 GeV)
and it decreases the differential cross-section at photon energies below a
certain fraction of the incident electron energy (dsigma/dk decreases
-4
significantly if k/T<=10 .)
Using the tabulated cross-section values of Seltzer and Berger together
with the Migdal corrected Bethe-Heitler formula we have computed
sigma(Z,T,k ) and we have used these computed values as ``data points''
c
in the fitting procedure. Calculating the ``low energy'' (T<=10Gev) data
we have applied the second Midgal correction to the results of Seltzer and
Berger. We have chosen the parameterizations:
and
where m is the mass of the electron,
with E= T+m. The constants xi , xi , alpha, beta, v , v are
sigma l sigma l
parameters to be fitted.
is the Midgal correction factor, with
2 2
The factors (T+m) / T(T+ 2m) and (T+ m) /(T+ 2m) come from the scaled
cross-section computed by Seltzer and Berger:
The functions F (Z,X,Y)(i =sigma, l) have the form
i
where F (X,Y) are polynomials of their variables X,Y
ij
F (X,Y) denotes in fact a function constructed from two polynomials
ij
where the polynomials P fulfil the conditions
ij
and we have performed a least-squares fit to determine the parameters.
The values of the parameters (xi , alpha, v , C for sigma and
sigma sigma i
Brem
xi , beta, V , C for E ) can be found in the DATA statement within the
l l Loss
functions GBRSGE and
GBRELE which compute the formula
(3) and (4) respectively.
The errors of the parameterizations (3) and (4) can be estimated as
We have performed a fit for the ``data'' without the Midgal corrections,
too. In this case we used the data of Seltzer and Berger without any
correction for T<=10GeV and we used the Bethe-Heitler cross-section for
T>=10Gev. The parameterized forms of the cross-section and energy loss
are the same as it was in the first fit (i.e. (3) and (4)), only the
numerical values of the parameters have changed. These values are in DATA
statements in the functions
GBRSGE and
GBRELE and this second kind of
parameterization can be activated using the
PATCHY switch +USE BETHE. (The
two parameterizations give different results for high electron energy.)
The energy loss due to soft photon Bremsstrahlung is tabulated at
initialization time as a function of the medium and of the energy by
routine GBRELA (see JMATE data
structure) from the following expression:
For a molecule or a mixture:
where w is the proportion by weight of the i th element. In the tables,
i
the dE/dx due to Bremsstrahlung is summed with the energy loss coming
from ionization.
The mean free path, lambda, for an electron to radiate a photon via
Bremsstrahlung is given by
where sum is the macroscopic cross-section (1/cm). This macroscopic
cross-section can be written as
and for a compound or mixture:
This mean free path is tabuled at initialization time as a function of the
medium and of the energy by routine
GBRSGA (see JMATE data
structure).
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The radiative e /e energy loss is:
Reference [bib-KIM1] says that:
``The differences between the radiative loss of positrons and electrons
are considerable and cannot be disregarded.
[...] The ratio of the radiative energy loss for positrons to that for
electrons obeys a simple scaling law, [...] is a function only of the
2
quantity T/Z ''
In other words:
The authors have calculated this function in the range
-7 2
10 <=((T)/(Z ))<=0.5 (here the kinetic energy T is expressed in MeV).
Their data can be fairly accurately reproduced using a parametrization:
where:
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This e /e energy loss difference is not a pure low-energy phenomenon (at
least for high Z), as it can be seen from Table
[more info].
Table: ratio of the e / e radiative energy loss in lead
(Z=82).
The scaling holds for the ratio of the total radiative energy losses, but
it is significantly broken for the photon spectrum in the screened case.
In case of a point Coulomb charge the scaling would hold also for the
spectrum. In other words:
If we consider the photon spectrum from Bremsstrahlung reported in [bib-KIM1] we see that:
Table: Difference in the energy loss and Bremsstrahlung
- + #
cross-section for e /e in Carbon and Lead with a cut for gamma and e of
0
10keV. DeltaE is the value without the correction for the difference
l
- +
e / e .
We further assume that:
In order to satisfy approximately the scaling law for the ratio of the
total radiative energy loss, we require for f(epsilon):
From the photon spectra we require:
We have choosen a simple function f:
Table: Difference in the energy loss and Bremsstrahlung
- + #
cross-section for e /e in Carbon and Lead with a cut for gamma and e of
0
1MeV. DeltaE is the value without the correction for the difference
l
- +
e / e .
Table: Difference in the shower development for e /e in
Carbon and Lead. No diff refers to the value without the correction for
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the difference e /e .
from the conditions (
[more info]), (
[more info]) we get:
We have defined weight factors F and F for the positron continuous
l sigma
energy loss and discrete Bremsstrahlung cross section:
where epsilon = ((k )/(T)) and k is the photon cut BCUTE. In this scheme
0 c c
the positron energy loss and discrete Bremsstrahlung can be calculated as:
As in this approximation the photon spectra are identical, the same
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SUBROUTINE is used for generating e /e Bremsstrahlung. The following
relations hold:
which is consistent with the spectra.
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The effect of this e /e Bremsstrahlung difference can be also seen in
e.m. shower development, when the primary energy is not too high. An
example can be found in table
[more info].
+--------------------------------------+
|CALL GBRSGA GBRSGA |
+--------------------------------------+
GBRSGA is called at initialization time by GPHYSI.
VALUE = GBRSGE(ZZ,T,BCUT)
VALUE = GBFSIG(T,C)
Method
k
Brem c
E (Z, T,k ) =int k((dsigma(Z,T,k))/ (dk))dk (1)
c 0
Loss
T
sigma (Z, T,k ) =int ((dsigma(Z, T,k))/(dk))dk (2)
Brem c k
c
Parameterization of energy loss and total cross-section
2 alpha
sigma(Z, T,k ) =((Z(Z+xi )(T+m) )/ (T(T+2m)))[ln(T/k )] F (Z, X,Y) (barn)(3)
c sigma c sigma
Brem 2 beta
E (Z, T,k ) =((Z(Z+xi )(T+m) )/ ((T+2m)))[((k C )/(T))] F (Z, X,Y) (GeV barn)(4)
c l c M l
Loss
X =ln(E/ m) , Y= ln(v E/k ) for the total cross-section sigma
sigma c
Brem
X =ln(T/ m) , Y= ln(k /v E) for the energy loss E
c l
Loss
2 2 2
C = ((1)/ (1+ ((nr lamb-a (T+m) )/(pik ))))
M 0
c
2 2
f(k/ T) = ((beta )/(Z ))k((dsigma)/(dk))
2 2
= ((T(T+2m))/ ((T+ m) Z ))k((dsigma)/(dk))
F (Z, X,Y) =F (X, Y)+ ZF (X,Y) (5)
i i i
0 1
5 5
F (X, Y) = (C + C X+ ... +C X ) +(C +C X +. ..+ C X )Y
i 1 2 6 7 8 12
0
5 2 5 5
+(C + C X+ ... +C X )Y +. .. +(C +C X +.. .+ C X )Y (6)
13 14 18 31 32 36
Y< =0
5 5
= (C + C X+ ... +C X ) +(C +C X +. ..+ C X )
1 2 6 7 8 12
5 2 5 5
+(C + C X+ ... +C X )Y +. .. +(C +C X +.. .+ C X )Y
37 38 42 55 56 60
Y>0
4 4
F (X, Y) = (C + C X+ ... +C X ) +(C +C X +. ..+ C X )Y
i 61 62 65 66 67 70
1
4 2 4 4
+(C + C X+ ... +C X )Y +. .. +(C +C X +.. .+ C X )Y (7)
71 72 75 81 82 85
Y< =0
4 4
= (C + C X+ ... +C X ) +(C +C X +. ..+ C X )Y
61 62 65 66 67 70
4 2 4
+(C + C X+ ... +C X )Y +. .. +(C +C X +.. .+ C X )Y
86 87 90 96 97 100 4
Y>0
(8)
neg
P (X, Y) for Y<=0 (9)
ij
pos
P (X, Y) for Y>0
ij
neg pos
P (X, Y= 0) = P (X,Y =0) (10)
ij ij
neg pos
(deltaP / deltaY) = (deltaP deltaY) (11)
Y=0 Y=0
ij ij
We have computed 4000 ``data points'' in the range
Z= 6;13;29; 47;74;92
10kev<=T< =10Tev
10kev<=k < =T
c
12-15% for T<=1MeV
((Deltasigma)/ (sigma)) = { <=5- 6% for 1MeVEnergy loss due to Bremsstrahlung
Brem
((dE)/ (dx))= ((Nrho)/(A))E (Z,T, k ) in GeV/cm (12)
c
Loss
((dE)/ (dx))= rhosum w ((1)/ (rho ))(dE/dx) (13)
i i i
Total cross-section of Bremsstrahlung
lambda =((1)/ (Sigma)) (14)
sum =((Nrhosigma(Z, E,k ))/(A)) (15)
c
sum =NrhoSigma p sigma(Z ,E,k )sum p A = Nrhosum (((w )/ (A )))sigma(Z ,E, k )(16)
i i i c i i i i i i i c
Corrections for e /e differences
# 2 2 #
- ((1)/(rho))(((dE)/(dx))) = ((N alphar )/(A))(T+ m)Z Phi (Z,T)
Av
rad e rad
# 2 2 T #
Phi (Z,T) = ((1)/ (alphar Z (T+ m)))int k((dsigma )/(dk))dk
0
rad e
+ - 2
eta = ((Phi (Z, T))/(Phi (Z,T))) = eta(((T)/(Z )))
rad rad
0 if x<=- 8
3 5
eta = { ((1)/ (2))+((1)/ (pi))arctan(a x +a x +a x ) if - 8
2
x = log (C((T)/ (Z )))(T in GeV)
6
C = 7. 5221x 10
a = 0. 415
1
a = 0. 0021
3
a = 0. 00054
5
+ - 2 + -
((Phi )/ (Phi ))= eta(((T)/(Z ))) ((((dsigma )/(dk)))/(((dsigma )/ (dk))))= does not scale
+ # + - + -
((dsigma )/ (dk))= S (((k)/(T))) ((S (k))/(S (k)))<=1 S (1)= 0 S (1)>0
+ -
((dsigma )/ (dk))= f(epsilon)((dsigma )/(dk)) epsilon =((k)/ (T))[phys340-1](17)
1
int f(epsilon)depsilon = eta[phys340-2]
f(0)= 1
. f(1)= 0 } for all Z,T[phys340-3] (19)
alpha
f(epsilon) = C(1- epsilon) C,alpha>0[phys340-4](20)
C = 1
alpha = ((1)/ (eta))-1 (alpha>0 because eta<1)
((1)/(eta))-1
f(epsilon) = (1- epsilon)
epsilon
0 1
F =((1)/ (epsilon )) int f(epsilon)depsilon F =((1)/(1- epsilon )) int f(epsilon)depsilon[phys340-5](21)
l 0 0 sigma 0 epsilon
0
+ - + -
(- ((dE)/(dx))) =F (- ((dE)/(dx))) sigma = F sigma
l Brems sigma Brems
((1)/(eta))-1
F = eta(1-epsilon ) )
0 l 0 sigma
((1)/(eta)) F >eta
) F = eta((1-(1- epsilon ) ))/(epsilon ))>eta((1-(1- epsilon ))/(epsilon )) =eta ) { l .
l 0 0 0 0 F