Muon Ionisation

The class G4MuIonisation provides the continuous energy loss due to ionisation and simulates the ‘discrete’ part of the ionisation, that is, delta rays produced by muons. Inside this class the following models are used:

  • G4BraggModel (valid for protons with \(T\) < 0.2 MeV)

  • G4BetheBlochModel (valid for protons with 0.2 MeV < T < 1 GeV)

  • G4MuBetheBlochModel (valid for protons with \(T\) > 1 GeV)

The limit energy 0.2 MeV is equivalent to the proton limit energy 2 MeV because of scaling relation (35), which allows simulation for muons with energy below 1 GeV in the same way as for point-like hadrons with spin 1/2 described in Mean Energy Loss.

For higher energies the G4MuBetheBlochModel is applied, in which leading radiative corrections are taken into account [SRKP97]. Simple analytical formula for the cross section, derived with the logarithmic are used. Calculation results appreciably differ from usual elastic \(\mu -e\) scattering in the region of high energy transfers \(m_e \ll T < T_{max}\) and give non-negligible correction to the total average energy loss of high-energy muons. The total cross section is written as following:

(155)\[\sigma (E,\epsilon ) = \sigma_{BB}(E, \epsilon ) \left[ 1 + \frac{\alpha}{2\pi} \ln\left( 1 + \frac{2\epsilon}{m_e} \right) \ln\left(\frac{4m_e E(E - \epsilon )}{m_{\mu}^2( 2\epsilon + m_e) }\right) \right] ,\]

here \(\sigma(E,\epsilon)\) is the differential cross sections, \(\sigma(E,\epsilon)_{BB}\) is the Bethe-Bloch cross section (177), \(m_e\) is the electron mass, \(m_{\mu}\) is the muon mass, \(E\) is the muon energy, \(\epsilon\) is the energy transfer, \(\epsilon = \omega + T\), where T is the electron kinetic energy and \(\omega\) is the energy of radiative gamma.

For computation of the truncated mean energy loss (31) the partial integration of the expression (155) is performed

\[S(E,\epsilon_{up}) = S_{BB}(E,\epsilon_{up}) + S_{RC}(E,\epsilon_{up}), \;\; \epsilon_{up} = \min(\epsilon_{max},\epsilon_{cut}),\]

where term \(S_{BB}\) is the Bethe-Bloch truncated energy loss (171) for the interval of energy transfer \((0 - \epsilon_{up})\) and term \(S_{RC}\) is a correction due to radiative effects. The function become smooth after log-substitution and is computed by numerical integration

\[S_{RC}(E,\epsilon_{up}) = \int^{\ln\epsilon_{up}}_{\ln\epsilon_1}\epsilon^2(\sigma(E,\epsilon) - \sigma_{BB}(E, \epsilon))d(\ln\epsilon),\]

where lower limit \(\epsilon_1\) does not effect result of integration in first order and in the class G4MuBetheBlochModel the default value \(\epsilon_1 = 100\) keV is used.

For computation of the discrete cross section (32) another substitution is used in order to perform numerical integration of a smooth function

\[\sigma(E) = \int^{1/\epsilon_{up}}_{1/\epsilon_{max}}\epsilon^2\sigma (E,\epsilon )d(1/\epsilon ).\]

The sampling of energy transfer is performed between \(1/\epsilon_{up}\) and \(1/\epsilon_{max}\) using rejection constant for the function \(\epsilon^2\sigma(E,\epsilon)\). After the successful sampling of the energy transfer, the direction of the scattered electron is generated with respect to the direction of the incident particle. The energy of radiative gamma is neglected. The azimuthal electron angle \(\phi\) is generated isotropically. The polar angle \(\theta\) is calculated from energy-momentum conservation. This information is used to calculate the energy and momentum of both scattered particles and to transform them into the global coordinate system.

Bibliography

SRKP97

R.P. Kokoulin S.R. Kelner and A.A. Petrukhin. Bremsstrahlung from muons scattered by atomic electrons. Physics of Atomic Nuclei, 60:576–583, April 1997.