+ -
+-------------+ +----------##
| Geant 3.10 | GEANT User's Guide | PHYS211 ##
+-------------+ +----------##
Author(s) : G.N.Patrick, L.Urban Submitted: 12.12.84
Origin : GEANT2 Revised: 17.12.92
+--------------------------------+
|CALL GPAIRG |
+--------------------------------+
+ - GPAIRG generates of e e -pair production by a photon. It uses a modified version of the random number techniques of Butcher and Messel [bib-BUTC] to sample the secondary electron/positron energies from the Coulomb corrected Bethe-Heitler differential cross-section. For the angular distribution of the pair, is calls the function GBTETH.
Input: via COMMON/GCTRAK/
Output: via COMMON/GCKING/
GPAIRG is called automatically from the tracking routine GTGAMA, if, and when, the parent photon reaches its decay 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 the e e -pair production by photons it gives the scaled angle for an electron (mass PARTM) of energy ENER which is EFRAC times the initial energy of the photon. GBTETH is called by GPAIRG.
We give a very short summary of the random number technique used here [bib-BUTC], [bib-HAMM]. The method is a combination of the composition and rejection Monte Carlo methods. Suppose we wish to sample x from the distribution f(x) and the (normalized) probability density function can be written as
n
f(x) =sum alpha f (x)g (x) (1)
i=1 i i i
where
alpha >0
i
f (x) density functions (normalized correctly)
i
0<=g (x)< =1
i
According to this method, x can be chosen by
It can be shown that this scheme is correct and the mean number of tries to accept a value is sum alpha . i i
In practice we have a good method of sampling from the distribution f(x), if
Thus the different possible decompositions of the distribution f(x) are not equivalent from the practical point of view (e.g. they can be very different in computational speed) and it can be very useful to make some kind of optimization. A remark of pratical importance is that if our distribution is not normalized (int f(x)dx= C>0;C '1), the method can be used in the same manner, the only difference is that the mean number of tries in this case is given by sum ((alpha )/(C)) . i i
The Bethe-Heitler differential cross-section with the Coulomb correction + - for a photon of energy E to produce a e e -pair one of which has an energy epsilonE (epsilon is the fraction of the photon energy transferred to one particle of the pair) is given as in [bib-EGS3]:
2 2 2 2
((dsigma(Z, E,epsilon))/(depsilon)) =((r alphaZ[Z+xi(Z)])/ (E )){[epsilon +(1-epsilon) ][Phi (delta)-F(Z)]+((2)/(3))epsilon(1- epsilon)[Phi (delta)-F(Z)]}(2)
1 2
0
where Phi (delta) are the screening functions depending on the screening i variable delta
1/3
delta =((136m)/ (Z E)) ((1)/(epsilon(1-epsilon))) m =electron mass
2
Phi (delta)= 20.867-3. 242delta+ 0.625delta
. 1 } delta<=1
2
Phi (delta)= 20.209-1. 930delta-0.086delta
2
Phi (delta) =Phi (delta)= 21.12-4. 184ln(delta+ 0.952) delta>1
1 2
4/ 3lnZ E<0. 05GeV
F(Z) ={ 4/ 3lnZ +4f (Z) . E> =0.05 GeV
c
2/3 1/3
xi(Z) =((ln (1440/Z ))/(ln (183/Z )- f (Z)))
c
f (Z) the Coulomb correction function
c
2 3
f (Z) = a(1/ (1+ a)+0. 20206-0.0369a +0. 0083*a -0. 002a )
c
2
a = (alpha* Z)
alpha = 1. /137.
The kinematical range for the variable epsilon is
((m)/ (E))<=epsilon< =1-((m)/(E)) (3)
The cross-section is symmetric with respect to the interchange of epsilon with 1-epsilon, so we can restrict epsilon to lie in the range epsilon = ((m)/(E))< =epsilon<=((1)/(2)) 0
After some algebraic manipulations we can decompose the cross-section as (note: the normalization is not important):
2
((dsigma)/ (depsilon))= sum alpha f (epsilon)g (epsilon) (4)
i=1 i i i
where 2 alpha = (((0.5-epsilon ) )/(3))F alpha = ((1)/(2))F 1 0 10 2 20
2 f (epsilon)= ((3)/((0.5- epsilon )))(epsilon-0.5) f (epsilon)= ((1)/(0.5- epsilon )) 1 0 2 0 g (epsilon)= F /F g (epsilon)= F /F 1 1 10 2 2 20 and F = F (delta)= 3Phi (delta)-Phi (delta)- 2F(Z) F = F (delta ) 1 1 1 2 10 1 min F = F (delta)= ((3)/(2))Phi (delta) +((1)/ (2))Phi (delta)-2F(Z) F = F (delta ) 2 2 1 2 20 2 min ((1)/(3)) delta = 4((136m)/(Z 3E)) min
delta is the minimal value of the screening variable delta. It can be min seen that the functions f (epsilon) are normalized and that the functions i
g (epsilon) are ``valid" rejection functions. i
Therefore, given a set of uniformly distributed random numbers r i (0<=r < =1), we can sample the variate epsilon (x in the program) by i
BPAR =((alpha )/ (alpha + alpha ))
1 1 2
If r It should be mentioned that we need a step just after sampling epsilon in the step 2, because the cross-section formula goes to negative values at large delta. The cross-section must not be allowed to go negative, so this imposes an upper limit for delta;
delta =exp [(21. 12-F(Z))/4. 184]-0.962
max
If we get a delta value using the sampled epsilon such that delta>delta , we have to start again from the step 1. After the max successful sampling of epsilon, GPAIRG generates the polar angles of the electron with respect to an axis defined along the direction of the parent photon. The electron and the positron are assumed to have a symmetric angular distribution. The energy-angle distribution is given by Tsai [bib-TSAI], [bib-TSAI-err] as following:
2 2 4 2 4 2
((dsigma)/ (dpdOmega)) = ((2alpha e )/(pikm )){[((2x(1- x))/((1+ l) ))-((12lx(1- x))/((1+ l) ))](Z + Z).
2 2 4 2 2
+.[((2x -2x +1)/ ((1+l) ))+((4lx(1- x))/((1 +l) ))][X-2Z f((alphaZ) )]}
(5)
where k is the photon energy, p and E are the momentum and the energy of + - 2 2 2 the electron of the e e -pair, x= E/k and 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 ) (6)
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 together with the momentum conservation is used to calculate the momentum vectors of both decay products and to transform them to the GEANT coordinate system. The choice of which particle in the pair is the electron/positron is made randomly.