Pair production¶
Models¶
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class BetheHeitlerInteractor¶
Relativistic model for electron-positron pair production.
The energies of the secondary electron and positron are sampled using the Bethe-Heitler cross sections with a Coulomb correction. The LPM effect is taken into account for incident gamma energies above 100 GeV. Exiting particle directions are sampled with the
TsaiUrbanDistribution. Note that energy is not exactly conserved.For additional context on the derivation see Butcher and Messel [1960] .
Note
This performs the same sampling routine as in Geant4’s G4PairProductionRelModel, as documented in sections 6.5 (gamma conversion) and 10.2.2 (LPM effect) of [The Geant4 Collaboration, 2023] (release 10.7)
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class MuPairProductionInteractor¶
Perform electron-positron pair production by muons.
Note
This performs the same sampling routine as in Geant4’s G4MuPairProductionModel and as documented in the Geant4 Physics Reference Manual (Release 11.1) section 11.3.
Distributions¶
The energy transfer for muon pair production is sampled using the inverse transform method with tabulated CDFs.
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class MuPPEnergyDistribution¶
Sample the electron and positron energies for muon pair production.
The energy transfer to the electron-positron pair is sampled using inverse transform sampling on a tabulated CDF. The CDF is calculated on a 2D grid, where the x-axis is the log of the incident muon energy and the y-axis is the log of the ratio of the energy transfer to the incident particle energy. Because the shape of the distribution depends only weakly on the atomic number, the CDF is calculated for a hardcoded set of points equally spaced in \( \log Z \) and linearly interpolated.
The formula used for the differential cross section is valid when the maximum energy transfer to the electron-positron pair lies between \( \epsilon_{\text{min}} = 4 m \), where \( m \) is the electron mass, and
where \( E = T + \mu \) is the total muon energy, \( \mu \) is the muon mass, \( e \) is Euler’s number, and \( Z \) is the atomic number.\[ \epsilon_{\text{max}} = E + \frac{3 \sqrt{e}}{4} \mu Z^{1/3}), \]The maximum energy partition between the electron and positron is calculated as
The partition \( r \) is then sampled uniformly in \( [-r_{\text{max}}, r_{\text{max}}) \).\[ r_{\text{max}} = \left[1 - 6 \frac{\mu^2}{E (E - \epsilon)} \right] \sqrt{1 - \epsilon_{\text{min}} / \epsilon}. \]