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Celeritas
0.5.0-86+4a8eea4
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Celeritas
0.5.0-86+4a8eea4
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Calculate the differential cross section for muon bremsstrahlung. More...
#include <MuBremsDiffXsCalculator.hh>
Public Types | |
Type aliases | |
| using | Energy = units::MevEnergy |
| using | Mass = units::MevMass |
Public Member Functions | |
| CELER_FUNCTION | MuBremsDiffXsCalculator (ElementView const &element, Energy inc_energy, Mass inc_mass, Mass electron_mass) |
| Construct with incident particle data and current element. | |
| CELER_FUNCTION real_type | operator() (Energy energy) |
| Compute the differential cross section per atom at the given photon energy. | |
Calculate the differential cross section for muon bremsstrahlung.
The differential cross section can be written as
\[ \difd{\sigma}{\epsilon} = \frac{16}{3} \alpha N_A (\frac{m}{\mu} r_e)^2 \frac{1}{\epsilon A} Z(Z \Phi_n + \Phi_e) (1 - v + \frac{3}{4} v^2), \]
where \( \epsilon \) is the photon energy, \( \alpha \) is the fine structure constant, \( N_A \) is Avogadro's number, \( m \) is the electron mass, \( \mu \) is the muon mass, \( r_e \) is the classical electron radius, \( Z \) is the atomic number, and \( A \) is the atomic mass. \( v = \epsilon / E \) is the relative energy transfer, where \( E \) is the total energy of the incident muon.
The contribution to the cross section from the nucleus is given by
\[ \Phi_n = \ln \frac{B Z^{-1/3} (\mu + \delta(D'_n \sqrt{e} - 2))}{D'_n (m + \delta \sqrt{e} B Z^{-1/3})} \ , \]
where \( \delta = \frac{\mu^2 v}{2(E - \epsilon)}\) is the minimum momentum transfer and \( D'_n \) is the correction to the nuclear form factor.
The contribution to the cross section from electrons is given by
\[ \Phi_e = \ln \frac{B' Z^{-2/3} \mu}{\left(1 + \frac{\delta \mu}{m^2 \sqrt{e}}\right)(m + \delta \sqrt{e} B' Z^{-2/3})} \ . \]
The constants \( B \) and \( B' \) were calculated using the Thomas-Fermi model. In the case of hydrogen, where the Thomas-Fermi model does not serve as a good approximation, the exact values of the constants were calculated analytically.
This performs the same calculation as in Geant4's G4MuBremsstrahlungModel::ComputeDMicroscopicCrossSection() and as described in section 11.2.1 of the Physics Reference Manual. The formulae are taken mainly from SR Kelner, RP Kokoulin, and AA Petrukhin. About cross section for high-energy muon bremsstrahlung. Technical Report, MEphI, 1995. Preprint MEPhI 024-95, Moscow, 1995, CERN SCAN-9510048.
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