+ -

Simulation of e e annihilation

+-------------+                                               +----------##
| Geant 3.16  |               GEANT User's Guide              | PHYS351  ##
+-------------+                                               +----------##
                                   

Author(s) : L.Urban Submitted: 26.10.84 Origin : Same Revised: 19.12.92

Subroutines

                     +------------------------------+
                     |CALL GANNI  |
                     +------------------------------+
                                  

GANNI generates the annihilation of a positron into two photons. It uses the following input and output:

input:
via common /GCTRAK/
output:
via common /GCKING/

The routine is called automatically from the tracking routine GTELEC, if the positron reaches its interaction point during the tracking.

                    +--------------------------------+
                    |CALL GANNIR  |
                    +--------------------------------+
                                  

GANNIR generates the positron annihilation at rest. It is called from the tracking routine GTELEC, if the positron energy is below the cut-off energy CUTELE in common block /GCCUTS/.

Method

The differential cross-section of the two-photon positron-electron annihilation can be written as [bib-HEIT], [bib-EGS3]:

    ((dsigma(Z, epsilon))/(depsilon))= ma[S(aepsilon)+S(a(1-epsilon))](1)
                                  

where m is the electron mass Z is the atomic number of the material and:

      gamma    =    ((E)/ (m))          with E the positron energy
          a    =    gamma +1
    epsilon    =    ((k)/ (E+m))       with k the energy of the secondary photon of the lower energy
                                               2
       S(x)    =    C [- 1+ ((C )/(x))-((1)/ (x ))]
                     1         2
                           2
           with  C  =((Zpir  )/(a(E-m)))  and  C  =a +((egamma)/ (a))
                  1                             2
                           0

The kinematical limits for the variable epsilon are given by

                                  2
    epsilon  =((1)/ (a+ sqrt(gamma -1)))< =epsilon<=((1)/ (2))         (2)
           0

Due to the symmetry of the formula (1) in epsilon, the range of epsilon can be expanded from (epsilon ,((1)/(2))) to (epsilon , 1-epsilon ) and 0 0 0 the second function S can be eliminated from the formula. Having done this, the differential cross-section can be decomposed (apart from the normalization) as

                                                                                     2         2                            2
    ((dsigma)/ (depsilon))= ((1)/(ln((1-epsilon )/ (epsilon ))))((1)/(epsilon)). (((a + 2a-2)-a epsilon- ((1)/(epsilon)))/(a - 2))(3)
                             ######################## #########################   ####################### ########################
                            |#######################{z########################}  |######################{z#######################}
                                                f(epsilon)                                          g(epsilon)

Using the expression (3) the secondary photon energy can be sampled by (r ,r random numbers). 0 1

  1. sampling epsilon from f(epsilon)
        epsilon =epsilon  exp [ln(((1-epsilon )/ (epsilon )))r ]           (4)
                        0                    0           0    0
    
  2. computing the rejection function g(epsilon) and if r <=g(epsilon) accepting epsilon 1 if r >g(epsilon) accepting epsilon, starting again from 1. 1

    After the successful sampling of epsilon, the photon energy is computed as

        k =(E +m)epsilon                                                   (5)
    

    and then GANNI generates the polar angles of the photon with respect to an axis defined by the momentum of the positron. The azimuthal angle Phi is generated isotropically and Theta is computed from the energy-momentum conservation. With this information, the momentum vector of both photons can be calculated and transformed into the GEANT coordinate system.

    The routine GANNIR treates the special case when a positron falls below the cut-off energy (CUTELE in common block /GCCUTS/) before annihilating. In this case, it is assumed that the positron comes to rest before annihilating (see the tracking routine GTELEC). GANNIR generates the photons, each with energy k= m. The angular distribution is isotropic, with the two photons going in opposite directions.

    Restrictions

    1. The annihilation processes producing one or three or more photons are ignored, because these processes are negligible compared to the annihilation into two photons [bib-EGS3], [bib-MES1]
    2. In calculating the process it is assumed that the atomic electron initially is free and at rest. This is the usual assumption used in shower programs [bib-EGS3].
    3. In the case of a compound or mixture target material, the effective atomic number (computed in GSMIXT) is used.