+ -
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| Geant 3.11 | GEANT User's Guide | PHYS450 ##
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Author(s) : L.Urban Submitted: 24.03.86
Origin : Same Revised: 19.12.92
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|CALL GPRELA |
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GPRELA fills the tables for the energy loss of muons due to the (soft) + - e e -pair production and nuclear interactions in different materials. For energy loss due pair production, it calls the function GPRELM. The energy binning is set within the array ELOW (COMMON CGMULO) in the routine GPHYSI. In the tables, the energy loss due to the pair production is summed with that from other processes. The following pointers are used:
GPRELA is called at initialization time by GPHYSI.
VALUE = GPRELM(Z,T,CUT)
GPRELM calculates the energy loss due to the direct pair production of a muon with kinetic energy T in material with atomic number Z. It is called by GPRELA for energies which are below the cut CUT. Above this cut, the direct pair production is simulated explicitly (see PHYS 451) and tabulation of these continuous losses is not needed. GPRELM is called by GPRELA.
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|CALL GPRSGA |
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GPRSGA calculates the total cross-section for the pair production of photons and the direct pair production of muons in all materials. It tabulates the mean free path, lambda= ((1)/(Sigma)) (in cm) as a function of medium and energy. For the direct pair production of muons, it calls the function GPRSGM. The energy binning is set within the array ELOW (COMMON CGMULO) in the routine GPHYSI. The following pointers are used:
GPRSGA is called at initialization time by GPHYSI.
VALUE = GPRSGM(Z,T,CCUT)
GPRSGM calculates the total cross-section for the direct pair production of a muon with kinetic energy T in material with atomic number Z. It is called by GPRSGA for kinetic energies which are above the cut CCUT and for which the pair production is simulated explicitly (see PHYS 451).
When a muon of high total energy E moves in the field of an atom of charge + - th + Z, it can radiate a e e -pair (4 order QED process). Let's call E and - E the total energy of the emitted positron and electron respectively. Defining
+ - + - + -
nu =((E +E )/ (E)) and rho= ((E -E )/ (E + E )) (1)
the differential cross-section for the process can be written [bib-LOHM]:
2 4 - 2 2
((d sigma)/ (dnudrho))= alpha ((2)/(3pi))(Zlambda) ((1-v)/ (v))[phi +(((m)/(M))) phi ](2)
e mu
where
The explicit form of the terms phi and phi can be found in HREF=H2PHYSBiblio.html#bib-LOHM>[bib-LOHM]. The kinematic ranges of nu and rho are:
1/3
((4m)/ (E))= nu <= nu < = nu = (1-3sqrt(e)((M)/ (4E))Z )
min max
(3)
2 2
0= rho <= |rho(nu)| < = rho (nu) = [1-((6M )/ (E (1-nu)))]sqrt(1- ((4m)/(nuE)))
min max
where e = 2.718
+ - E is the total energy cut-off below which the emitted e e -pair are c treated as continuous energy loss by the muon, and above which they are explicitly generated (PPCUTM in the program), and v = E /E. The mean c c
+ - value of the energy lost by the incident muon due to the (soft) e e -pair is:
nu rho (nu)
c max 2
(Z, T,E ) =2E nut dnu nu int d rho((d sigma)/(dnudrho)) (GeV barn /atom)(4)
soft c min 0
+ - whereas the total cross-section for the emission of a hard e e -pair is:
nu rho (nu)
max max 2
sigma(Z, T,E ) =2 int dnunu int drho((d sigma)/ (dnudrho)) (barn / atom)(5)
c nu 0
c
2
Instead of using the explicit formula for d sigma/dnudrho (1) we have
chosen to parameterize directly
where xi , xi , alpha, beta, gamma and delta are parameters obtained
sigma l
min
by a fitting procedure and E = Ev =4m and E = Ev
c min e max max
The functions F (Z,X,Y) (i= sigma, l) have the form:
i
where F (X,Y) denotes a function constructed from two polynomials
ij
neg pos
and P , P fulfil the conditions:
ij ij
The detailed form of the P polynomials are:
ij
with
By a numerical (twofold) integration of the formulae (2), (6) and (7)
above, we have calculated 1800 ``data points" in the range Z= 1, 6, 13,
26, 50, 82, 92 and the energy ranges 1GeV <=T< = 10 TeV and 4m<=E < =T
c
and performed a least-squares fit to determine the parameters. The fitted
values of the parameters are in the DATA statements in the functions
GPRSGM and
GPRELM, which compute the
formulae (9) and (8) respectively. The accuracy of the fit:
10% for T< =5Gev
((Deltasigma)/(sigma))<= { 5% for T>5Gev .
((Delta
The function GPRELM contains a
second formula to calculate the total energy lost by the muon due to
+ -
direct e e -production (this formula is used if
1/3
E >=E =E- 0.75pem Z ). This formula gives the total energy loss
c max mu
with an error less than 1%. It has the form:
1/3
where X= ln(E/M) and X= Z . The fitted parameters d can be found in
i
the DATA statement within the function
GPRELM.
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The energy lost by the muon due to the (soft) e e -pair radiated is
tabulated at initialization time as a function of the medium and of the
energy by the routine GPRELA.
For a mixture or a molecule:
+ -
In the table, the dE/dx due to the e e -pair production is summed with
the energy loss coming from others processes, namely ionization,
Bremsstrahlung and nuclear interaction.
+ -
The mean free path, lambda, for a muon to radiate a e e -pair is given
by:
where Sigma is the macroscopic cross-section (1/cm). This macroscopic
cross-section can be written as:
for a compound or mixture:
This mean free path is tabulated at initialization time as a function of
the medium and of the energy by routine
GPRSGA.
alpha
sigma(Z, T,E ) = Z[Z +xi (1 +gammaln Z)][ln((E )/(E))] F (Z, X(6)
c sigma max sigma
min beta
F (Z, X,Y) =F (X, Y)+ZF (X,Y) (8)
i i0 i1
neg
P (X,Y) if Y< =0
F (X, Y)= { pos . (9)
ij P (X,Y) if 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
neg 5
P (X, Y) = (C + C X+ ... +C X ) +(C +C X +. ..+ C X )Y
1 2 6 5 7 8 12
i0
5 2 5 5
+(C + C X+ ... +C X )Y +. ..+ (C + .. .+C X )Y
13 14 18 31 36
pos 5 5
P (X, Y) = (C + C X+ ... +C X ) +(C +C X +. ..+ C X )Y
1 2 6 7 8 12
i0
5 2 5 5
+(C + C X+ ... +C X )Y +. ..+ (C + .. .+C X (12)
37 38 42 55 60
neg 4 4
P (X, Y) = (C + C X+ ... +C X ) +(C +C X +. ..+ C X )Y
i1 61 62 65 66 67 70
4 2 4 4
+(C + C X+ ... +C X )Y +. ..+ (C + .. .+C X )Y
71 72 75 81 85
pos 4 4
P (X, Y) = (C + C X+ ... +C X ) +(C +C X +. ..+ C X )Y
61 62 65 66 67 70
i1
4 2 4 4
+(C + C X+ ... +C X )Y +. ..+ (C + .. .+C X )Y
86 87 90 96 100
pair
Energy loss due to the direct pair production
((dE)/ (dx))= ((Nrho)/(A))
((dE)/ (dx))= rhosum w ((1)/ (rho ))(((dE)/(dx))) (16)
i i i
where w is the proportion by weight of the i'th element.
i
Total cross-section of the direct pair production
lambda =((1)/ (Sigma)) (in cm) (17)
Sigma =((Nrhosigma(Z, E,E ))/(A)) (18a)
c
Sigma =((Nrho sum p sigma(Z , E,E ))/(sum p A )) (18b)
i i i c i i i