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| Geant 3.16 | GEANT User's Guide | PHYS430 ##
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Author(s) : G.N.Patrick, L.Urban, D.Ward Submitted: 12.03.82
Origin : Geant 2 Revised: 19.12.92
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|CALL GDRELA |
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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. To compute the energy loss of protons due to the ionization, GDRELP is called, and the value is stored in the energy loss table. This value is used for other charged hadrons with scaled energy bin. For muons, GDRELM is called, and the energy loss due to the ionization is summed with the energy loss from the other processes (i.e. Bremsstrahlung, pair production and nuclear interaction). The following pointers are used:
JMA = LQ(JMATE-I) pointer to the I'th material
+ -
JEL1 = LQ(JMA-1) pointer for dE/ dx for e +e -
JEL2 = LQ(JMA-2) pointer for dE/ dx for mu /mu
JEL3 = LQ(JMA-3) pointer for dE/ dx for all others particles
GDRELA is called at initialization time by GPHYSI.
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| CALL GDRELP (A,Z,DENS,T,DEDX) |
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GDRELP computes the ionization energy loss (DEDX) for protons with kinetic energy T in the medium with the atomic weight A, the atomic number Z and the density DENS. It is called by the routine GDRELA.
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| CALL GDRELM (A,Z,DENS,T,DEDX) |
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GDRELM computes the ionization energy loss (DEDX) for muons with kinetic energy T in the medium with the atomic weight A, the atomic number Z and the density DENS. It is called by routine GDRELA.
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|CALL GDRSGA |
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CDRSGA calculates the total - - cross-section in all materials for delta rays for Moeller (e e ) and
- + Bhabha (e e ) scattering and for muons. For hadrons, this value is calculated at tracking time. CDRSGA 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:
JMA = LQ(JMATE-I) pointer to the I'th material
JDRAY = LQ(JMA-11) pointer to Delta ray cross-section bank
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's call dsigma(E,T)/dT the differential cross-section for the ejection of an electron with kinetic energy T by a charged particle of total energy E moving in a medium of density rho. T (in the program DCUTM) is the cut kinetic energy cut-off below which the soft emitted electrons are treated as continuous energy lost by the incident particle, and above which they are explicitly generated.
The mean value of the energy lost by the incident particle due to the ejected soft electrons is:
T
EL(E, T )= inttT((dsigma(E, T))/(dT))dT (1)
cut 0
whereas the total cross-section for the ejection of an electron of energy T>T is: cut
T
max
sigma(E, T )= int ((dsigma(E,T))/ (dT))dT (2)
cut T
cut
where T is the maximum energy transferable to the free electron: max
2 2
T =((2m(gamma -1))/(1 +2gamma((m)/(M)) +(((m)/ (M))) )) (3)
max
where
In this chapter, the method of calculation of the continuous energy loss and the total cross-section are explained. The explicit generation of delta rays is explained in chapter PHYS 331.
The integration of (1) leads to the restricted energy loss formula [bib-PDGB]:
2 2 2
((1)/ (rho))(((dE)/(dx)))= D((Z)/(beta ))[((1)/ (2))ln(((T TC)/(I )))- ((beta )/(2))-((delta)/ (2))-((C )/(Z))](4)
max e
where,
I is the average ionization potential of the atom in question. There exists a variety of phenomenological approximations for this, and the following quoted by Bricman et al. [bib-BRIC] has been adopted here
0.9
I =16(Z) eV (5)
This somewhat arbitrary choice does not, however, represent a serious source of error since I only enters into the logarithmic term of the equation.
delta is a correction term which takes account of the reduction in energy loss due to the so-called density effect. This becomes important at high energy because media have a tendency to become polarized as the incident particle velocity increases. As a consequence, the atoms in a medium can no longer be considered as isolated. To correct for this effect the formulation of Sternheimer [bib-STE1], [bib-STE2] is used:
= 0 X =X
1
2 where, 4.606X= ln(gamma -1)
X ,X ,a, m and C are constants which depend on the medium and are 0 1 calculated as below:
C =-2ln(I/ hnu )-1
p
where
a m
a =((4. 606(X -X ))/ ((X -X ) ))
0 1 0
where, X =-C/ 4.606 and: a
when I<100eV: X = 2.0,m = 3.0 i)X =0. 2 for - C<3.681 0 ii)X =- 0.326C-1. 0 for - C>3.681 0 when I>=100eV : X = 3.0,m = 3.0 1 i)X =0. 2 for - C<5.215 0 ii)X =- 0.326C-1. 5 for - C>5.215 0
C /Z is a shell correction term which accounts for the fact that at low e energies for light elements or at all energies for heavy elements the probability of particle-electron collisions at deep electronic layers (K, L, etc.) is negligible. Barkas [bib-BARK] has published a semi-empirical formula which is applicable to all materials, and this is utilized:
-2 -4 -6 2 -2 -4 -6 -9 3
C (I, eta)= (0.42237eta +0.0304eta -0.00038eta- 610 I + (3.858eta -0. 1668eta + 0.00158eta )10 I
e
where eta= betagamma. (C itself is a dimensionless constant, but as I in e the original article was expressed in eV and in GEANT it is expressed in
2 2 -6+9 12 GeV, the exponent of ten in the I -term is 10 = 10 , and that of the 3 3 -9+9 18 I -term is 10 = 10 .) However, this formula breaks down at very low energies and is thus only applied if eta>0.13.
GDRELP has been tested against energy loss tables [bib-SERR], [bib-SER1] for various materials in the energy range 0-25 GeV. Typical discrepancies are as follows:
Beryllium: 1.1% at 0.05 GeV 0.02% at 25 GeV Hydrogen : 1.5% at 0.05 GeV 12.1% at 25 GeV Water : 8.1% at 0.05 GeV 4.4% at 6 GeV
The energy lost due to the soft delta rays is tabulated at initialization time as a function of the medium and of the energy by routine GDRELA The tables are filled with the quantity dE/dx in GeV/cm (formula 4 above) and for a molecule or a mixture:
((dE)/ (dx))= rhosui W ((1)/ (pi ))(((dE)/(dX))) (7)
i i i
where W is the proportion by weight of the ith element. i
The energy loss of all charged particles can be obtained from that of protons by calculating the equivalent proton kinetic energy of the particle and searching for the correct energy bin in the table:
T =((M )/(M))T (8)
proton p
The integration of formula [2] gives the total cross-section :
2 2 2
spin-0 particle: sigma(Z,E, T ) = 2pir m((Z)/(beta ))((1)/ (T ))(1-Y+ beta Yln Y) (9)
cut 0 cut
2 2 2
spin-((1)/ (2)) particle: sigma(Z,E, T ) = sigma(Z,E,T ) +2pir m((Z)/ (beta ))(((T -T )/ (2E(10)
cut cut max cut
0
where Y= T /T < =1 cut max
The macroscopic cross-section sum (in 1/cm) is:
Sigma = ((Nrhosigma(Z, E,T ))/(A))
cut
i
Sigma = ((Nrho sim p sigma(Z ,E, T ))/(sui p A ))= Nrhosui (((w )/ (A )))*sigma(Z ,E, T(11)
i cut i i i i i cut
for the chemical element or compound/mixture respectively, where
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 for leptons only.
The cross-section (9,10) is strongly dependent of the mass of the incident particle, and cannot be tabulated in a general way for any charged hadrons. Therefore, for such particles, the cross-section is computed at tracking time in the routine GTHADR.