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| Geant 3.16 | GEANT User's Guide | PHYS440 ##
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Author(s) : L.Urban Submitted: 26.10.84
Origin : Same Revised: 19.12.92
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|CALL GBRELA |
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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 muons, GBRELA calls the function GDRELM. The following pointers are used:
GBRELA is called at initialization time by GPHYSI.
VALUE = GBRELM(Z,T,BCUT)
GBRELM calculates the energy loss due to Bremsstrahlung of a muon with kinetic energy T in material with atomic number Z. It is called by GBRELA for energies which are below the cut BCUT. Above this cut, the Bremsstrahlung process is simulated explicitly (see PHYS 441) and tabulation of these continuous losses is not needed. GBRELM is called by GBRELA.
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|CALL GBRSGA |
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GBRSGA 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 the medium and the energy. The energy binning is set within the array ELOW (COMMON CGMULO) in the routine GPHYSI. The following pointers are used:
GBRSGA is called at initialization time by GPHYSI.
VALUE = GBRSGM(Z,T,BCUT)
GBRSGM calculates the total cross-section of Bremsstrahlung of a muon with kinetic energy T in material with atomic number Z. It is called by GBRSGA for kinetic energies which are above the cut BCUT and for which Bremsstrahlung process is simulated explicitly (see PHYS 441).
Let us call dsigma(Z,T,k)/dk the differential cross-section for the production of a photon of energy k by a muon of kinetic energy T in the field of an atom of charge Z. k is the energy cut-off below which the c soft photons are treated as a continuous energy lost by the muon, and above which they are explicitly generated (BCUTM in the program.)
Then the mean value of the energy lost by the muon due to soft photons is:
k
c
EL(Z, T,k ) =int k((dsigma(Z, T,k))/(dk))dk (1)
c 0
whereas the total cross-section for the emission of a photon of energy >k is c
T
sigma(Z, T,k ) =int ((dsigma(Z,T, k))/(dk))dk (2)
c k
c
The most accurate cross-section formula for the high energy (T>=1 GeV) muon Bremsstrahlung can be found in a paper of W. Lohmann and R. Voss [bib-LOHM].
We have used the cross-section from [bib-LOHM] to calculate ``data" points sigma(Z,T,k ) and EL(Z, T,k ). We have parameterized the cross-section c c and energy loss as:
alpha
sigma(Z, T,k ) = Z[Z+ xi (1+ gammaln Z)][ln((k )/(k ))] F (Z, X,Y) in b(3)
c sigma max c sigma
beta
EL(Z, T,k ) = Z[Z+ xi (1+ deltaln Z)][ln((k )/(E))] F (Z, X,Y) in GeV barn (4)
c l c l
1/3 where k = E-0.75sqrt(e)m Z is the maximum possible value of the max mu
photon energy,
The functions F (Z,X,Y)(i =sigma, l) are polynomials of their variables i
5 5
F (Z, X,Y) = (C +C X +.. .+ C X )+ (C + C X+ ... +C X )Y
i 1 2 6 7 8 12
5 5
+. ..+ (C + C X+ .. .+C X )Y +
31 32 36
3 3
+Z[(C +C X +.. .+ C X )+ (C + C X+ ... +C X )Y
37 38 40 41 42 44
3 3
+. ..+ (C + C X+ .. .+C X )Y )] (5)
48 49 52
We have computed more than 2000 ``data" points in the range: Z = 1,6,13,26,50,82,92 for 1 GeV <=T< = 10 TeV and 10 keV <=k < =T. We c performed a least-squares fit to determine the values of the parameters in formulae (3-5). These values (xi , gamma, alpha ,C for sigma and sigma i xi , delta, beta, C for EL) can be found in the DATA statements within l i the functions GBRSGM and GBRELM which compute the formulae (3) and (4) respectively.
The accuracy of the fit can be estimated as:
((Deltasigma)/(sigma)) 10- 12% for T< =5GeV < =4% for T>5GeV ((DeltaEL)/(EL)) 10% for T< =5GeV < =3% for T>5GeV
(But the contribution of the Bremsstrahlung to the total energy loss of the muons is less than 1% for T<= 5 GeV.)
We have made a separate parameterization for the total energy lost by the muon due to Bremsstrahlung. If k <=k we have chosen the following c max parameterization:
brem
EL (Z, T)= EL(Z,T,k = k )=
max
Z(Z +1)k [d +(d X +d Y) (6)
max 1 2 3
2 2 6 5 6
+(d X +d XY +d Y )+. .. +(d X +d X Y +.. .+ d Y )]
4 5 6 22 23 28
1/3
k = E- 0.75sqrt(e)m Z
max mu
X = ln(E/ m )
p
1/3
Y = Z
E = T +m
mu
The accuracy of the formula (6) is rather good,
brem brem
((DeltaEL )/ (EL )) <= 1. 5% if T>1GeVand 1< =Z<=100
brem brem
((DeltaEL )/ (EL )) <= 1% if T>5GeV
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.
((dE)/ (dx))= ((Nrho)/(A))EL(Z,T, k ) in GeV/cm (7)
c
For a molecule or a mixture:
((dE)/ (dx))= rhoSigmaw ((1)/(rho ))(((dE)/ (dx))) (8)
i i i
where w is the proportion by weight of the i'th element. i
In the tables, the dE/dx due to Bremsstrahlung is summed with the energy lost coming from ionization, pair production and nuclear interaction.
lambda =1/ Sigma (9)
where Sigma is the macroscopic cross-section (1/cm). This macroscopic cross-section can be written as
Sigma =((Nrhosigma(Z, E,k ))/(A)) (10)
c
for a compound or mixture:
Sigma = ((Nrho sum (Z ,E,k ))/ (sum p A )) =Nrho sum (((w )/(A ))). sigma(Z ,E(11)
i i c i i i i i i i c
This mean free path is tabuled at initialization time as a function of the medium and of the energy by routine GBRSGA.