Simulation of the delta-ray production

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| Geant 3.15  |               GEANT User's Guide              | PHYS331  ##
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Author(s) : L.Urban, D.Ward                            Submitted: 26.10.84
Origin    : L.Urban                                      Revised: 17.12.92
                                  

Subroutines

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                     |CALL GDRAY  |
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GDRAY generates explicitly the delta-rays (see PHYS330 for treatment of the ionization as continuous energy cross and for the calculation of the total cross-section).

input:

via /GCTRAK/

output:

via /GCKING/

The routine is called from the tracking routines GTELEC, GTMUON and GTHADR when a charged particle reaches its interaction point.

Method

Differential cross-section

The differential cross-section of the delta-ray production can be written as (1,2) [bib-MES1], [bib-EGS3], [bib-PDGB]:

                                   2        2                    2        2                                                       2                                                           2
    ((dsigma)/ (depsilon))= ((2piZr m)/(beta (E- m)))[(((gamma-1) )/(gamma ))+((1)/ (epsilon))(((1)/(epsilon))-((2gamma- 1)/(gamma )))+((1)/ (1-epsilon))(((1)/(1- epsilon))((2gamma-1)/(gamma )))](1)
                                   0

for the electron/electron (Moeller) scattering and

                                 2                       2       2                                           2
    ((dsigma)/ (depsilon))((2piZr m)/((E-m)))[((1)/ (beta epsilon ))-((B )/ (epsilon))+B -B epsilon+B epsilon ](2)
                                                                        1               2  3         4
                                 0

for the positron-electron (Bhabha) scattering. In these equations,

  Z          =    the atomic number of the medium
  E          =    the energy of the incident particle
  M          =    the rest mass of the incident particle
  gamm2      =    ((E)/(M))    2
  beta       =    1-((1)/(gamma ))
  y          =    ((12/(gamma+ 1))
  B          =    2-y
   1
                             2
  B          =    (1-2y)(3+ y )
   2                    2        3             3
  B          =    (1-2y) + (1-2y) , B  =(1- 2y)
   3                                 4
  epsilon    =    ((T)/(E-m))

with T the kinematic energy of the scattered electron (of the lower energy - + in the case of e e scattering).

The kinematical limits for the variable epsilon are:

         -  +
    for e  e     :    epsilon = ((TCUT)/(E-m))< =epsilon< =((1)/(2))   (3)
                             0
         +  -
    for e  e     :    epsilon = ((TCUT)/(E-m))< =epsilon< =1           (4)
                             0

For the other charged particles the differential cross-section can be written:

                                  2           2         2          2
    ((dsigma)/ (dT))    =    2piZr m((1)/(beta ))((1)/(T  ))[1-beta ((T)/(TMAX))] for spin 0 particle              (5)
                                  0
                                  2           2         2          2                 2      2
    ((dsigma)/ (dT))    =    2piZr m((1)/(beta ))((1)/(T  ))[1-beta ((T)/(TMAX)) +((T )/ (2E ))] for spin 1/2 parti(6)
                                  0

where TMAX is the maximum energy transferable to the free electron:

                    2                             2
    TMAX =((2m(gamma - 1))/(1 +2gamma(m/M) +(m/ M) ))                  (7)

and TCUT is the energy threshold for the delta-ray emission: TCUT<=T< =TMAX

Sampling

Apart from the normalization, the cross-section can be written as

    ((dsigma)/ (depsilon))= f(epsilon)g(epsilon),                      (8)

- - where, for e e scattering,

                                                               2
    f(epsilon)    =    ((epsilon )/ (1-2epsilon ))((1)/(epsilon  ))                                                                        (9)
                                0              o
                                    2                       2       2         2                                            2              2
    g(epsilon)    =    ((4)/ (9gamma -10gamma+ 5))[(gamma-1) epsilon - (2gamma + 2gamma-1)((epsilon)/(1- epsilon))+ ((gamma )/((1-epsilon)(10)

+ - and for the case of the e e scattering

                                                              2
    f(epsilon)    =    ((epsilon )/ (1-epsilon ))((1)/(epsilon  ))                                                          (11)
                                0             0
                                                 2          3           4                            2           3           4
    g(epsilon)    =    ((B - B epsilon+ B epsilon -B epsilon  +B epsilon )/ (B -B epsilon  +B epsilon - B epsilon + B epsilo(12)
                          0   1          2          3           4             0  1       0   2           3           4
                                                                                                     0           0           0

2 2 Here B = gamma /(gamma - 1), all the other quantities have been defined 0 above.

For the other charged particles:

    f(T)    =    (((1)/ (TCUT))-((1)/(TMAX))). ((1)/(T))                                      (13)
                        2                 2     2
    g(T)    =    1- beta ((T)/(TMAX))+ ((T )/(2E ))  (last term for spin-((1)/ (2)) particle o(14)

GDRAY samples the variate epsilon by:

1. sampling epsilon from f(epsilon)
2. calculating the rejection function g(epsilon) and accepting the sampled epsilon with a probability of g(epsilon).