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Subsections

Gamma Conversion into an Electron - Positron Pair

Cross Section and Mean Free Path

Cross Section per Atom

The total cross-section per atom for the conversion of a gamma into an pair has been parameterized as

 (6.14)

where is the incident gamma energy and . The functions are given by
 (6.15)

with the parameters taken from a least-squares fit to the data [1]. Their values can be found in the function which computes formula 6.14.
This parameterization describes the data in the range

and

The accuracy of the fit was estimated to be with a mean value of . Above 100 GeV the cross section is constant. Below the extrapolation
 (6.16)

is used.

Mean Free Path

In a given material the mean free path, , for a photon to convert into an pair is
 (6.17)

where is the number of atoms per volume of the element of the material.

Final State

Choosing an Element

The differential cross section depends on the atomic number of the material in which the interaction occurs. In a compound material the element in which the interaction occurs is chosen randomly according to the probability
 (6.18)

Corrected Bethe-Heitler Cross Section

As written in [2], the Bethe-Heitler formula corrected for various effects is
 (6.19)

where is the fine-structure constant and the classical electron radius. Here , is the energy of the photon and is the total energy carried by one particle of the pair. The kinematical limits of are therefore
 (6.20)

Screening Effect

The screening variable, , is a function of
 (6.21)

and measures the 'impact parameter' of the projectile. Two screening functions are introduced in the Bethe-Heitler formula :
 (6.22)

Because the formula 6.19 is symmetric under the exchange , the range of can be restricted to
 (6.23)

Born Approximation

The Bethe-Heitler formula is calculated with plane waves, but Coulomb waves should be used instead. To correct for this, a Coulomb correction function is introduced in the Bethe-Heitler formula :
 (6.24)

with
 (6.25)

It should be mentioned that, after these additions, the cross section becomes negative if
 (6.26)

This gives an additional constraint on :
 (6.27)

where
 (6.28)

has been introduced. Finally the range of becomes
 (6.29)

Gamma Conversion in the Electron Field

The electron cloud gives an additional contribution to pair creation, proportional to (instead of ). This is taken into account through the expression
 (6.30)

Factorization of the Cross Section

is sampled using the techniques of 'composition+rejection', as treated in [3,4,5]. First, two auxiliary screening functions should be introduced:
 (6.31)

It can be seen that and are decreasing functions of , . They reach their maximum for :
 (6.32)

After some algebraic manipulations the formula 6.19 can be written :
 (6.33)

where

and are probability density functions on the interval such that

, and and are valid rejection functions: .

Sampling the Energy

Given a triplet of uniformly distributed random numbers :
1. use to choose which decomposition term in 6.33 to use:
 (6.34)

2. sample from or with :
 (6.35)

3. reject if
NOTE : below MeV it is enough to sample uniformly on , without rejection.

Charge

The charge of each particle of the pair is fixed randomly.

Polar Angle of the Electron or Positron

The polar angle of the electron (or positron) is defined with respect to the direction of the parent photon. The energy-angle distribution given by Tsai [6] is quite complicated to sample and can be approximated by a density function suggested by Urban [7] :
 (6.36)

with
 (6.37)

A sampling of the distribution 6.36 requires a triplet of random numbers such that
 (6.38)

The azimuthal angle is generated isotropically.

Final State

The and momenta are assumed to be coplanar with the parent photon. This information, together with energy conservation, is used to calculate the momentum vectors of the pair and to rotate them to the global reference system.

Status of this document

12.01.02 created by M.Maire.
21.03.02 corrections in angular distribution (mma)
22.04.02 re-worded by D.H. Wright

Bibliography

1. J.H.Hubbell, H.A.Gimm, I.Overbo Jou. Phys. Chem. Ref. Data 9:1023 (1980)
2. W. Heitler The Quantum Theory of Radiation, Oxford University Press (1957)
3. R. Ford and W. Nelson. SLAC-210, UC-32 (1978)
4. J.C. Butcher and H. Messel. Nucl. Phys. 20 15 (1960)
5. H. Messel and D. Crawford. Electron-Photon shower distribution, Pergamon Press (1970)
6. Y. S. Tsai, Rev. Mod. Phys. 46 815 (1974), Y. S. Tsai, Rev. Mod. Phys. 49 421 (1977)
7. L.Urban in GEANT3 writeup, section PHYS-211. Cern Program Library (1993)

Next: Gamma Conversion into a Up: Gamma Incident Previous: Compton scattering   Contents