[1208] | 1 | \chapter{Lepton Hadron Interactions} |
---|
| 2 | |
---|
| 3 | The photonuclear interaction of muons is currently the only process treated |
---|
| 4 | in this category. |
---|
| 5 | |
---|
| 6 | \section{\it G4MuonNucleusProcess} |
---|
| 7 | |
---|
| 8 | This class simulates the photonuclear interaction of muons in a material. |
---|
| 9 | The muon interacts electromagnetically with a nucleus, exchanging a virtual |
---|
| 10 | photon. At energies above a few GeV, the photon interacts hadronically with |
---|
| 11 | the nucleus, producing hadronic secondaries. |
---|
| 12 | |
---|
| 13 | The outcome of the simulation depends heavily upon the interaction model |
---|
| 14 | chosen. Hence the model-dependent part of the process is implemented in |
---|
| 15 | the {\it G4MuonNucleusInteractionModel} class, which can be easily replaced by |
---|
| 16 | another model. |
---|
| 17 | |
---|
| 18 | {\it G4MuonNucleusInteractionModel} calculates the cross section and final |
---|
| 19 | states of the muon and hadronic secondaries. The final muon momentum is |
---|
| 20 | given by a double-differential cross section which depends on the |
---|
| 21 | photoabsorption cross sections for longitudinally and transversely polarized |
---|
| 22 | photons. The final hadronic state is determined by replacing the virtual |
---|
| 23 | photon with a charged pion of the same $Q^2$ and then allowing the pion to |
---|
| 24 | interact with the nucleus. The charge of the pion is chosen at random. The |
---|
| 25 | pion interactions with the nucleus are modeled by processes derived from the |
---|
| 26 | GHEISHA \cite{GHEISHA} package. These processes are: \\ |
---|
| 27 | |
---|
| 28 | \begin{tabular}[t]{ll} |
---|
| 29 | {\it G4LEPionPlusInelastic}, {\it G4LEPionMinusInelastic} & $E \leq 25$ GeV \\ |
---|
| 30 | {\it G4HEPionPlusInelastic}, {\it G4HEPionMinusInelastic} & $E > 25$ GeV \\ |
---|
| 31 | \end{tabular} |
---|
| 32 | \\ |
---|
| 33 | |
---|
| 34 | \subsection{Cross Section Calculation} |
---|
| 35 | |
---|
| 36 | The cross section for the above process in a material is given roughly by |
---|
| 37 | |
---|
| 38 | \begin{eqnarray*} |
---|
| 39 | \sigma_{\mu A} = A \sigma_{\mu N} |
---|
| 40 | \end{eqnarray*} |
---|
| 41 | |
---|
| 42 | \noindent where $A$ is the atomic mass number of the material and |
---|
| 43 | $\sigma_{\mu N}$ is the cross section for the process on a single nucleon: |
---|
| 44 | |
---|
| 45 | \[ |
---|
| 46 | \sigma_{\mu N} = |
---|
| 47 | \left\{ \begin{array}{ll} |
---|
| 48 | 0.3 & (E \leq 30GeV) \\ |
---|
| 49 | 0.3 (E/30)^{0.25} & (E > 30GeV) \\ |
---|
| 50 | \end{array} \right. [\mu b] . |
---|
| 51 | \] |
---|
| 52 | |
---|
| 53 | \noindent |
---|
| 54 | The differential cross section, in terms of muon energy $E$ and emission solid |
---|
| 55 | angle $\Omega$, can be expressed as: |
---|
| 56 | |
---|
| 57 | \begin{eqnarray*} |
---|
| 58 | \frac{d\sigma}{d\Omega dE} =\Gamma\,(\sigma_T + \epsilon \sigma_L) |
---|
| 59 | \end{eqnarray*} |
---|
| 60 | where $\sigma_L$ and $\sigma_T$ are the photoabsorption cross sections for |
---|
| 61 | longitudinal and transverse photons, respectively. $\Gamma$ is the transverse |
---|
| 62 | photon flux and $\epsilon$ is the polarization of the intermediate photon. |
---|
| 63 | The photoabsorption cross sections are parameterized as: |
---|
| 64 | \begin{eqnarray*} |
---|
| 65 | \sigma_L &=& 0.3\,\left( 1 - \frac{1}{1.868} Q^2 \nu \right)\,\sigma_T \\ |
---|
| 66 | \sigma_T &\sim& const = 0.12 mb \\ |
---|
| 67 | \end{eqnarray*} |
---|
| 68 | |
---|
| 69 | \noindent while the flux and polarization are given by |
---|
| 70 | \begin{eqnarray*} |
---|
| 71 | \Gamma &=& \frac{K \alpha}{2\pi} \frac{E^\prime}{E} \frac{1}{1-\epsilon} \\ |
---|
| 72 | \epsilon &=& \left[ 1 + 2 \frac{Q^2 + \nu^2}{Q^2} tan^2 \frac{\theta}{2} \right]^{-1} . \\ |
---|
| 73 | \end{eqnarray*} |
---|
| 74 | |
---|
| 75 | \noindent |
---|
| 76 | $E$ and $E^{\prime}$ are the initial and final muon energies, $Q^2$ and $\nu$ |
---|
| 77 | are the scaling variables |
---|
| 78 | \begin{eqnarray*} |
---|
| 79 | Q^2 &=& -q^2 = 2 (EE^{\prime} - PP^{\prime} cos \theta - m_\mu^2) \\ |
---|
| 80 | \nu &=& E - E^{\prime} , \\ |
---|
| 81 | \end{eqnarray*} |
---|
| 82 | and $K$ is given using the Gilman convention |
---|
| 83 | \begin{eqnarray*} |
---|
| 84 | K = \nu + \frac{Q^2}{2\nu} . |
---|
| 85 | \end{eqnarray*} |
---|
| 86 | |
---|
| 87 | |
---|
| 88 | \section{Status of this document} |
---|
| 89 | 20.04.02 re-written by D.H. Wright \\ |
---|
| 90 | 23.10.98 created by M.Takahata \\ |
---|
| 91 | |
---|
| 92 | \begin{latexonly} |
---|
| 93 | |
---|
| 94 | \begin{thebibliography}{99} |
---|
| 95 | \bibitem{GHEISHA} H.Fesefeldt |
---|
| 96 | {\em GHEISHA The Simulation of Hadronic Showers} 149 |
---|
| 97 | {\em RWTH/PITHA 85/02} (1985) |
---|
| 98 | \end{thebibliography} |
---|
| 99 | |
---|
| 100 | \end{latexonly} |
---|
| 101 | |
---|
| 102 | \begin{htmlonly} |
---|
| 103 | |
---|
| 104 | \section{Bibliography} |
---|
| 105 | |
---|
| 106 | \begin{enumerate} |
---|
| 107 | \item H.Fesefeldt |
---|
| 108 | {\em GHEISHA The Simulation of Hadronic Showers} 149 |
---|
| 109 | {\em RWTH/PITHA 85/02} (1985) |
---|
| 110 | \end{enumerate} |
---|
| 111 | |
---|
| 112 | \end{htmlonly} |
---|
| 113 | |
---|