[1211] | 1 | \section[Ionization]{Muon Ionization} |
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| 2 | |
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| 3 | |
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| 4 | The class {\it G4MuIonisation} provides the continuous energy loss due to |
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| 5 | ionization and simulates the 'discrete' part of the ionization, that is, |
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| 6 | delta rays produced by muons. |
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| 7 | Inside this class the following models are used: |
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| 8 | \begin{itemize} |
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| 9 | \item |
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| 10 | {\it G4BraggModel} (valid for protons with $T < 0.2 \;MeV$) |
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| 11 | \item |
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| 12 | {\it G4BetherBlochModel} (valid for protons with $0.2 \;MeV < T < 1\; GeV$) |
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| 13 | \item |
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| 14 | {\it G4MuBetherBlochModel} (valid for protons with $T > 1\; GeV$) |
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| 15 | \end{itemize} |
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| 16 | The limit energy $0.2\; MeV$ is equivalent to the proton limit energy $2 MeV$ because of |
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| 17 | scaling relation (\ref{enloss.sc}), which allows |
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| 18 | simulation for muons with energy below $1\; GeV$ |
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| 19 | in the same way as for point-like hadrons with spin 1/2 described in the section \ref{en_loss}. |
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| 20 | |
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| 21 | For higher energies the {\it G4MuBetherBlochModel} is applied, in which |
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| 22 | leading radiative corrections are taken into account \cite{muion.br}. |
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| 23 | Simple analytical formula for the cross section, derived with the logarithmic |
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| 24 | are used. Calculation results appreciably differ from usual elastic $\mu -e$ |
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| 25 | scattering in the region of high energy transfers $m_e << T < T_{max}$ |
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| 26 | and give non-negligible correction to the total average energy loss of |
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| 27 | high-energy muons. The total cross section is written as following: |
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| 28 | \begin{equation} |
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| 29 | \label{muion.c} |
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| 30 | \sigma (E,\epsilon ) = \sigma_{BB}(E, \epsilon ) \left[ 1 + \frac{\alpha}{2\pi} |
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| 31 | ln\left( 1 + \frac{2\epsilon}{m_e} \right) |
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| 32 | ln\left(\frac{4m_e E(E - \epsilon )}{m_{\mu}^2( 2\epsilon + m_e) }\right) \right] , |
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| 33 | \end{equation} |
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| 34 | here $\sigma(E,\epsilon)$ is the differential cross sections, |
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| 35 | $\sigma(E,\epsilon)_{BB}$ is the Bethe-Bloch cross section (\ref{hion.i}), |
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| 36 | $m_e$ is the electron mass, $m_{\mu}$ is the muon mass, |
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| 37 | $E$ is the muon energy, $\epsilon$ |
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| 38 | is the energy transfer, $\epsilon = \omega + T$, where |
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| 39 | T is the electron kinetic energy and $\omega$ is the energy of radiative gamma. |
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| 40 | |
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| 41 | For computation of the truncated mean energy loss (\ref{comion.a}) the partial integration |
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| 42 | of the expression (\ref{muion.c}) is performed |
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| 43 | \begin{equation} |
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| 44 | \label{muion.d} |
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| 45 | S(E,\epsilon_{up}) = S_{BB}(E,\epsilon_{up}) + S_{RC}(E,\epsilon_{up}), \;\; \epsilon_{up} = min(\epsilon_{max},\epsilon_{cut}), |
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| 46 | \end{equation} |
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| 47 | where term $S_{BB}$ is the Bethe-Bloch truncated energy loss (\ref{hion.d}) for the interval |
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| 48 | of energy transfer $(0 - \epsilon_{up})$ |
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| 49 | and term $S_{RC}$ is a correction due to radiative effects. |
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| 50 | The function become smooth after log-substitution and |
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| 51 | is computed by numerical integration |
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| 52 | \begin{equation} |
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| 53 | \label{muion.e} |
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| 54 | S_{RC}(E,\epsilon_{up}) = |
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| 55 | \int^{\ln\epsilon_{up}}_{\ln\epsilon_1}\epsilon^2(\sigma(E,\epsilon) - \sigma_{BB}(E, \epsilon))d(\ln\epsilon), |
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| 56 | \end{equation} |
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| 57 | where lower limit $\epsilon_1$ does not |
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| 58 | effect result of integration in first order and |
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| 59 | in the class {\it G4MuBetheBlochModel} the default value $\epsilon_1 = 100 keV$ is used. |
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| 60 | |
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| 61 | For computation of the discrete cross section (\ref{comion.b}) another substitution is used |
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| 62 | in order to perform numerical integration of a smooth function |
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| 63 | \begin{equation} |
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| 64 | \label{muion.f} |
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| 65 | \sigma(E) = \int^{1/\epsilon_{up}}_{1/\epsilon_{max}}\epsilon^2\sigma (E,\epsilon )d(1/\epsilon ). |
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| 66 | \end{equation} |
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| 67 | The sampling of energy transfer is performed between $1/\epsilon_{up}$ and $1/\epsilon_{max}$ using |
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| 68 | rejection constant for the function $\epsilon^2\sigma(E,\epsilon)$. |
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| 69 | After the successful sampling of the energy transfer, the direction |
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| 70 | of the scattered electron is generated with respect to the direction of the |
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| 71 | incident particle. The energy of radiative gamma is neglected. |
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| 72 | The azimuthal electron angle $\phi$ is generated isotropically. |
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| 73 | The polar angle $\theta$ is calculated from energy-momentum conservation. |
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| 74 | This information is used to calculate the energy and momentum of both |
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| 75 | scattered particles and to transform them into the {\em global} coordinate |
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| 76 | system. |
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| 77 | |
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| 78 | \subsection{Status of this document} |
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| 79 | 09.10.98 created by L. Urb\'an. \\ |
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| 80 | 14.12.01 revised by M.Maire \\ |
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| 81 | 30.11.02 re-worded by D.H. Wright \\ |
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| 82 | 01.12.03 revised by V. Ivanchenko \\ |
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| 83 | 22.06.07 rewritten by V. Ivanchenko \\ |
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| 84 | |
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| 85 | \begin{latexonly} |
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| 86 | |
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| 87 | \begin{thebibliography}{99} |
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| 88 | |
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| 89 | \bibitem{muion.br} |
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| 90 | S.R.~Kelner, R.P.~Kokoulin, A.A.~Petrukhin, |
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| 91 | Phys. Atomic Nuclei 60 (1997) 576. |
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| 92 | \end{thebibliography} |
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| 93 | |
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| 94 | \end{latexonly} |
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| 95 | |
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| 96 | \begin{htmlonly} |
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| 97 | |
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| 98 | \subsection{Bibliography} |
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| 99 | |
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| 100 | \begin{enumerate} |
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| 101 | \item |
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| 102 | S.R.~Kelner, R.P.~Kokoulin, A.A.~Petrukhin, |
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| 103 | Phys. Atomic Nuclei 60 (1997) 576. |
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| 104 | \end{enumerate} |
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| 105 | |
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| 106 | \end{htmlonly} |
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