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| Geant 3.16 | GEANT User's Guide | PHYS441 ##
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Author(s) : L. Urban Submitted: 26.10. 84
Origin : Same Revised: 19.12.92
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|CALL GBREMM |
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GBREMM generates a photon from Bremsstrahlung of a highly energetic muon by treating it as a discrete process. For the angular distribution of the photon, is calls the function GBTETH.
Input : via COMMON/GCTRAK/
Output: via COMMON/GCKING/
GBREMM is called automaticailly from the tracking routine GTMUON if, and when, the muon reaches its radiation point during the tracking stage of GEANT.
VALUE = GBTETH(ENER,PARTM,EFRAC)
+ - GBTETH calculates the angular distribution for the e e -pair in pair production and for the photon in Bremsstrahlung. In case of Bremsstrahlung it gives the scaled angle for a photon radiated by a muon (mass PARTM) of energy ENER. The energy of photon is EFRAC times the initial energy of the muon. GBTETH is called by GBREMM.
The differential cross-section of a photon of energy k by a muon of energy E is [bib-LOHM], [bib-MARM]:
2
((dsigma)/ (dv))= N((1)/(v))(((4)/(3))- ((4)/(3))v+v )Theta(delta)(1)
where N = normalization factor (not important here)
v = ((k)/ (E))
2
delta = ((M )/ (2E))((v)/(1-v)) minimum momentum transfer to the nucleus
M = muon mass
m = electron mass
e = 2. 718
1/3 1/3
Phi(delta) = ln(((189M)/ (mZ )))-ln(((189sqrt(e))/(mZ ))delta +1) if Z< =10
1/3
Phi(delta) = Phi(delta) +ln(((2)/(3Z ))) if Z>10
and
1/3
v = ((kc)/ (E))<=v< =(1-((3sqrt(e)MZ )/(4E))) = V
c MAX
Therefore, the differential cross-section can be written as
((dsigma)/ (dv))# f(v)g(v) (2)
with
-1
f(v) = [v ln(((V )/(vc)))]
MAX
2
g(v) = ((1)/ (Phi(0)))(1-v+ ((3v )/(4)))Phi(delta)
We can sample the photon energy in the following way (r , r uniformly 1 2 distributed random numbers [0,1]):
After the successful sampling of k, GBREMM generates the polar angles of the radiated photon with respect to an axis defined along the parent muon's momentum. The energy-angle distribution is given by Tsai [4] as follows:
2 2 4 2 4 2
((dsigma)/ (dkdOmega)) = ((2alpha e )/(pikm )){[((2y- 2)/((1+ l) ))+ ((12l(1-y))/((1 +l) ))](Z +Z).
2 2 4 2 2
+.[((2-2y- y )/((1+ l) ))-((4l(1- y))/((1+ l) ))][X-2Z f((alphaZ) )]}
(3)
where k is the photon energy, E is the initial muon energy, y= k/E, 2 2 2 l= E theta /m . This distribution is quite complicated for sampling and, furthermore, for a variable u= Etheta/m, shows a very weak dependence on Z, E(k), y= k/E. Thus, the distribution can be approximated by a function
-au -3au
f(u) =C(ue +due ) (4)
where
2
C = ((9a )/ (9+ d))
a = 0. 625
d = 27. 0
The sampling of the function f(u) can be done in the following way (r i
are uniformly distributed random numbers in [0,1]):
The azimuthal angle, Phi, is generated isotropically. This information is used to calculate the momentum vector of the radiated photon, to transform it to the GEANT coordinate system and to store the result into COMMON /GCKING/. Also, the momentum of the parent muon is updated.