Index: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/electromagnetic/standard/MuPgen.tex
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+\section[Gamma Conversion into a Muon - Anti-mu Pair]
+ {Gamma Conversion into a Muon - Anti-muon Pair}
+
+The class {\tt G4GammaConversionToMuons} simulates the process of gamma 
+conversion into muon pairs.  Given the photon energy and $Z$ and $A$ of the 
+material in which the photon converts, the probability for the conversions 
+to take place is calculated according to a parameterized total cross section.
+Next, the sharing of the photon energy between the $\mu^+$ and $\mu^-$ is
+determined.  Finally, the directions of the muons are generated.  Details of
+the implementation are given below and can be also found in\,\cite{MuPgen}.
+
+\subsection{Cross Section and Energy Sharing}
+In the field of the nucleus, muon pair production on atomic electrons,
+$\gamma+e\to e+\mu^+ +\mu^-$, has a threshold of
+$2m_\mu(m_\mu+m_e)/m_e\approx 43.9\;{\rm GeV}$ .
+%hbu check in MuonBkg.C  in fact 43.905 GeV
+Up to several hundred GeV this process has a much lower cross section than 
+the corresponding process on the nucleus.  At higher energies, the cross 
+section on atomic electrons represents a correction of $\sim 1/Z$ to the 
+total cross section.
+
+For the approximately elastic scattering considered here, momentum, but no 
+energy, is transferred to the nucleon.  The photon energy is fully shared 
+by the two muons according to
+\begin{equation}
+E_\gamma = E_\mu^+ + E_\mu^-
+\end{equation}
+or in terms of energy fractions
+\[ x_+ = \frac{E_\mu^+}{E_\gamma}, \qquad x_- = \frac{E_\mu^-}{E_\gamma}, \qquad x_+ + x_- = 1\;.
+\]
+The differential cross section for electromagnetic pair creation of muons in
+terms of the energy fractions of the muons is
+\begin{equation}
+\frac{d\sigma}{d x_+} = 4 \, \alpha \, Z^2 \, r_c^2
+\left(1-\frac43\,x_+x_-\right)\log(W)\;,
+\label{eq:dSigxPlus}
+\end{equation}
+where $Z$ is the charge of the nucleus, $r_c$ is the classical radius of the
+particles which are pair produced (here muons) and
+\begin{equation}
+W = W_\infty \;
+\frac{1+(D_n\sqrt{e}-2)\,\delta\,/m_\mu}{1+B \, Z^{-1/3}\, \sqrt e \,\delta\,/m_e}
+\label{eq:W}
+\end{equation}
+where
+\[
+W_\infty = \frac{B \, Z^{-1/3}}{D_n} \, \frac{m_\mu}{m_e} \qquad
+\delta = \frac{m_\mu^2}{2\,E_\gamma\,x_+x_-} \qquad 
+\sqrt{e}=1.6487\dots .
+\]
+\begin{eqnarray}
+\mbox{For hydrogen} \qquad B=202.4 \qquad D_n=&1.49 & \nonumber\\
+\mbox{and for all other nuclei} \qquad B=183 \qquad D_n=&1.54 \, A^{0.27}. & 
+\end{eqnarray}
+These formulae are obtained from the differential cross section for muon 
+bremsstrahlung \cite{Kelner:1995hu} by means of crossing relations.  The
+formulae take into account the screening of the field of the nucleus by the
+atomic electrons in the Thomas-Fermi model, as well as the finite size of 
+the nucleus, which is essential for the problem under consideration.
+%
+The above parameterization gives good results for $E_\gamma \gg m_\mu$.  The
+fact that it is approximate close to threshold is of little practical 
+importance.  Close to threshold, the cross section is small and the few low 
+energy muons produced will not travel very far.  The cross section 
+calculated from Eq.\,(\ref{eq:dSigxPlus}) is positive for 
+$E_\gamma > 4 m_\mu$ and
+\begin{equation}
+x_{\rm min} \leq x \leq x_{\rm max} \quad {\rm with} \quad
+x_{\rm min} = \frac12 - \sqrt{\frac{1}{4}-\frac{m_\mu}{E_\gamma}} \qquad
+x_{\rm max} = \frac12 + \sqrt{\frac{1}{4}-\frac{m_\mu}{E_\gamma}}\;,
+\end{equation}
+except for very asymmetric pair-production, close to threshold, which can 
+easily be taken care of by explicitly setting $\sigma = 0$ whenever 
+$\sigma < 0$.
+
+Note that the differential cross section is symmetric in $x_+$ and $x_-$ and
+that
+\[ x_+ x_- = x - x^2 \] 
+where $x$ stands for either $x_+$ or $x_-$.  By defining a constant
+\begin{equation}
+\sigma_0 = 4 \, \alpha \, Z^2 \, r_c^2 \log(W_\infty)\;
+\label{eq:sigma0}
+\end{equation}
+the differential cross section Eq.\,(\ref{eq:dSigxPlus}) can be rewritten 
+as a normalized and symmetric as function of $x$: 
+\begin{equation}
+\frac{1}{\sigma_0} \, \frac{d\sigma}{dx} = \left[ 1-\frac43 \,(x - x^2) \right]
+\, \frac{\log W}{\log W_\infty}\;.
+\label{eq:dSigdx}
+\end{equation}
+This is shown in Fig.\,\ref{plot:dsigdx} for several elements and a wide 
+range of photon energies.  The asymptotic differential cross section for 
+$E_\gamma \rightarrow \infty$
+\[  \frac{1}{\sigma_0} \, \frac{d\sigma_\infty}{dx} = 1-\frac43 \,(x - x^2) \]
+is also shown.
+\begin{figure}[htpb]
+\center\includegraphics[scale=.65]{electromagnetic/standard/MuPgen/dsigdx.eps}
+\caption{Normalized differential cross section for pair production as a 
+function of $x$, the energy fraction of the photon energy carried by one of 
+the leptons in the pair.  The function is shown for three different 
+elements, hydrogen, beryllium and lead, and for a wide range of photon 
+energies.}
+\label{plot:dsigdx}
+\end{figure}
+
+\subsection{Parameterization of the Total Cross Section}
+The total cross section is obtained by integration of the differential 
+cross section Eq.\,(\ref{eq:dSigxPlus}), that is
+\begin{equation}
+\sigma_{\rm tot}(E_\gamma) = \int_{x_{\rm min}}^{x_{\rm max}} \frac{d\sigma}{d x_+} \, d x_+
+= 4 \, \alpha \, Z^2 \, r_c^2 \, \int_{x_{\rm min}}^{x_{\rm max}}\left(1-\frac43\,x_+x_-\right)\log(W) \, d x_+ \;.
+\label{eq:sigmatot}
+\end{equation}
+$W$ is a function of ($x_+, E_\gamma$) and ($Z, A$) of the element 
+(see Eq.\,(\ref{eq:W})).  Numerical values of $W$ are given in 
+Table\,\ref{tab:W}.
+
+\begin{table}[htbp]\center
+\caption{Numerical values of $W$ for $x_+=0.5$ for different elements.}
+\label{tab:W}\vskip 1mm
+\begin{tabular}{|c|c|c|c|c|} \hline
+$E_\gamma$ & W for H & W for Be & W for Cu & W for Pb \\
+ GeV      & & & & \\ \hline
+     1    & 2.11  & 1.594  & 1.3505 & 5.212 \\
+    10    & 19.4  & 10.85  & 6.803 & 43.53  \\
+   100    & 191.5 & 102.3  & 60.10 & 332.7  \\
+  1000    & 1803  & 919.3  & 493.3 & 1476.1 \\
+ 10000    & 11427 & 4671   & 1824  & 1028.1 \\
+ $\infty$ & 28087 & 8549   & 2607  & 1339.8 \\ \hline
+\end{tabular}
+\end{table}
+%
+Values of the total cross section obtained by numerical integration are 
+listed in Table\,\ref{tab:sigmatot} for four different elements.  Units are 
+in $\mu{\rm barn}\,$, where $1\,\mu{\rm barn} = 10^{-34}\,{\rm m}^2\,$.
+%
+\begin{table}[htbp]\center
+\caption{Numerical values for the total cross section}
+\label{tab:sigmatot}\vskip 1mm
+\begin{tabular}{|c|c|c|c|c|} \hline
+$E_\gamma$ & $\sigma_{\rm tot}$, H & $\sigma_{\rm tot}$, Be & $\sigma_{\rm tot}$, Cu & $\sigma_{\rm tot}$, Pb \\
+ GeV      & $\mu{\rm barn}\,$       & $\mu{\rm barn}\,$       & $\mu{\rm barn}\,$       & $\mu{\rm barn}\,$      \\ \hline
+     1    & 0.01559 & 0.1515 & 5.047 & 30.22 \\
+    10    & 0.09720 & 1.209  & 49.56 & 334.6 \\
+   100    & 0.1921  & 2.660  & 121.7 & 886.4 \\
+  1000    & 0.2873  & 4.155  & 197.6 & 1476  \\
+ 10000    & 0.3715  & 5.392  & 253.7 & 1880  \\
+ $\infty$ & 0.4319  & 6.108  & 279.0 & 2042  \\ \hline
+\end{tabular}
+\end{table}
+%
+\begin{figure}[htpb]
+\center\includegraphics[scale=.7]{electromagnetic/standard/MuPgen/SigTot.eps}
+\caption{Total cross section for the Bethe-Heitler process 
+$\gamma \rightarrow \mu^+\mu^-$ as a function of the photon energy 
+$E_\gamma$ in hydrogen and lead, normalized to the asymptotic cross section 
+$\sigma_\infty$.}
+\label{plot:SigTot}
+\end{figure}
+
+\noindent
+Well above threshold, the total cross section rises about linearly in 
+$\log(E_\gamma)$ with the slope
+\begin{equation}
+W_M = \frac{1}{4\, D_n \, \sqrt e \,m_\mu}
+\end{equation}
+until it saturates due to screening at $\sigma_\infty$.  
+Fig.\,\ref{plot:SigTot} shows the normalized cross section where
+\begin{equation}
+\sigma_\infty = \frac79 \, \sigma_0 \qquad {\rm and} \qquad \sigma_0 = 4 \, \alpha \, Z^2 \, r_c^2 \, \log(W_\infty)\;.
+\end{equation}
+Numerical values of $W_M$ are listed in Table\,\ref{tab:WM}.
+
+\begin{table}[htbp]\center
+\caption{Numerical values of $W_M$.}
+\label{tab:WM}\vskip 1mm
+\begin{tabular}{|c|c|} \hline
+Element & $W_M$ \\
+        & 1/GeV \\ \hline
+H       & 0.963169 \\
+Be      & 0.514712 \\
+Cu      & 0.303763 \\
+Pb      & 0.220771 \\
+\hline
+\end{tabular}
+\end{table}
+
+
+The total cross section can be parameterized as
+\begin{equation}
+\sigma_{\rm par} = \frac{28 \, \alpha \, Z^2 \, r_c^2}{9}  \; \log(1 + W_M C_f E_g)\;,
+\label{eq:sigpar}
+\end{equation}
+with
+\begin{equation}
+E_g = \left(1-\frac{4 m_\mu}{E_\gamma}\right)^{t}
+\left(W_{\rm sat}^{s} + E_\gamma^{s} \right)^{1/s}\;.
+\end{equation}
+and
+\[
+W_{\rm sat} = \frac{W_\infty}{W_M} = B \, Z^{-1/3} \, \frac{4\,\sqrt e \,m_\mu^2 }{m_e}\;.
+\]
+The threshold behavior in the cross section was found to be well 
+approximated by $t = 1.479 + 0.00799 D_n $ and the saturation by
+$ s = -0.88 $.  The agreement at lower energies is improved using an 
+empirical correction factor, applied to the slope $W_M$, of the form
+\[
+C_f = \left[ 1 + 0.04 \log \left(1+\frac{E_c}{E_\gamma}\right)\right]\;,
+\]
+where
+\[ E_c = \left[ -18.+\frac{4347.}{B \, Z^{-1/3}}\right] \;{\rm GeV}\;.
+\]
+A comparison of the parameterized cross section with the numerical 
+integration of the exact cross section shows that the accuracy of the 
+parametrization is better than 2\%, as seen in Fig.\,\ref{plot:SigApRat}.
+\begin{figure}[htpb]
+\center\includegraphics[scale=.8333]{electromagnetic/standard/MuPgen/SigApRat.eps}
+\caption{Ratio of numerically integrated and parametrized total cross 
+sections as a function of $E_\gamma$ for hydrogen, beryllium, copper and 
+lead.}
+\label{plot:SigApRat}
+\end{figure}
+
+\subsection{Multi-differential Cross Section and Angular Variables}
+The angular distributions are based on the multi-differential cross section
+for lepton pair production in the field of the Coulomb center
+\[
+\frac{d\sigma}{dx_+ \, du_+ \, du_-\,d\varphi} = \frac{4\,Z^2\alpha^3}{\pi}\,\frac{m_\mu^2}{q^4}\,u_+\,u_-
+\]
+\[
+\left\{ \frac{u_+^2+u_-^2}{(1+u_+^2)\,(1+u_-^2)} -2x_+x_- \right.
+\]
+\begin{equation}
+\left.
+\left[\frac{u_+^2}{(1+u_+^2)^2}+\frac{u_-^2}{(1+u_-^2)^2}\right]
+-\frac{2u_+u_-(1-2x_+x_-)\,\cos\varphi}{(1+u_+^2)\,(1+u_-^2)}
+\right\} \,. \\
+\label{eq:MultiDiff}
+\end{equation}
+Here
+\begin{equation}
+u_\pm = \gamma_\pm \theta_\pm \quad , \qquad \gamma_\pm = \frac{E_\mu^\pm}{m_\mu}
+\quad,\qquad q^2=q_{\parallel}^2+q_{\perp}^2\quad, %new
+\label{eq:uthetagamma}
+\end{equation}
+where
+\begin{eqnarray}
+  q_{\parallel}^2=q_{\min}^2\,(1+x_-u_+^2+x_+u_-^2)^2\,, \nonumber \\
+  q_{\perp}^2=m_\mu^2\left[(u_+-u_-)^2+2\,u_+u_-(1-\cos\varphi)
+  \right]\,. \,
+\label{eq:q2}
+\end{eqnarray}
+$q^2$ is the square of the momentum ${\bf q}$ transferred to the target 
+and $q_{\parallel}^2$ and $q_{\perp}^2$ are the squares of the components 
+of the vector ${\bf q}$, which are parallel and perpendicular to the 
+initial photon momentum, respectively.
+The minimum momentum transfer is 
+$q_{\min}=m_\mu^2/(2E_\gamma \, x_+x_-)$.\\
+
+The muon vectors have the components
+\begin{equation}
+  \begin{array}{rcl}\displaystyle
+  {\bf p}_+&=&p_+\,(\;\;\;\sin\theta_+\cos(\varphi_0+\varphi/2)\,,\,\;\;\;\sin\theta_+\sin(\varphi_0+\varphi/2)\,,\,\cos\theta_+)\,,\\ 
+  {\bf p}_-&=&p_-\,(     -\sin\theta_-\cos(\varphi_0-\varphi/2)\,,\,     -\sin\theta_-\sin(\varphi_0-\varphi/2)\,,\,\cos\theta_-)\,,
+  \end{array}
+\label{eq:pvec}
+\end{equation}
+where $p_{\pm}=\sqrt{E_{\pm}^2-m_\mu^2}$.
+The initial photon direction is taken as the $z$-axis.
+The cross section of Eq.\,(\ref{eq:MultiDiff}) does not depend on 
+$\varphi_0$.  Because of azimuthal symmetry, $\varphi_0$ can simply be 
+sampled at random in the interval $(0,\,2\,\pi)$.
+
+Eq.\,(\ref{eq:MultiDiff}) is too complicated for efficient Monte Carlo 
+generation.  To simplify, the cross section is rewritten to be symmetric
+in $u_+$, $u_-$ using a new variable $u$ and small parameters $\xi,\beta$, 
+where $u_\pm=u \pm \xi/2$ and $\beta = u \,\varphi$.  When higher powers 
+in small parameters are dropped, the differential cross section in terms 
+of $u,\xi,\beta$ becomes
+\begin{eqnarray}
+\label{mupgen.a}
+ \frac{d\sigma}{dx_+ \, d\xi \, d\beta\, u du} & = & \frac{4\,Z^2\alpha^3}{\pi}
+\frac{m_\mu^2}{\left(q_{\parallel}^2+m_\mu^2(\xi^2+\beta^2)\right)^2} \\
+& &   \left\{\xi^2\left[\frac1{(1+u^2)^2}-2\,x_+x_-\,\frac{(1-u^2)^2}{(1+u^2)^4}\right]+
+  \frac{\beta^2(1-2x_+x_-)}{(1+u^2)^2}\right\}\,, \nonumber
+\label{eq:MultiDiff2}
+\end{eqnarray}
+where, in this approximation,
+$$
+q_{\parallel}^2=q_{\min}^2\,(1+u^2)^2\,.\,
+$$
+For Monte Carlo generation, it is convenient to replace ($\xi,\beta$) by 
+the polar coordinates ($\rho,\psi$) with $\xi=\rho\,\cos\psi$ and 
+$\beta=\rho\,\sin\psi$.  Integrating Eq.~\ref{mupgen.a} over $\psi$ and 
+using symbolically $du^2$ where $du^2 = 2 u \, du$ yields
+\begin{equation}
+\label{mupgen.b}
+\frac{d\sigma}{dx_+\,d\rho\,du^2} =\frac{4Z^2\alpha^3}{m_\mu^2}\,\frac{\rho^3}{(q_{\parallel}^2/m_\mu^2+\rho^2)^2}
+\,\,\left\{\frac{1-x_+x_-}{(1+u^2)^2}-\frac{x_+x_-(1-u^2)^2}{(1+u^2)^4}\right\}.\,
+\end{equation}
+Integration with logarithmic accuracy over $\rho$ gives
+\begin{equation}\label{q4}
+\int\!\frac{\rho^3\,d\rho}{(q_{\parallel}^2/m_\mu^2+\rho^2)^2}
+\approx \int\limits_{q_{\parallel}/m_\mu}^1\!\frac{d\rho}{\rho}
+=\log\left(\frac{m_\mu}{q_{\parallel}}\right)\,.
+\end{equation}
+Within the logarithmic accuracy, $\log(m_\mu/q_{\parallel})$ can be 
+replaced by $\log(m_\mu/q_{\min})$, so that
+\begin{equation}
+\frac{d\sigma}{dx_+\,du^2}=\frac{4\,Z^2\alpha^3}{m_\mu^2}\,
+\left\{\frac{1-x_+x_-}{(1+u^2)^2}-\frac{x_+x_-(1-u^2)^2}{(1+u^2)^4}\right\}\,
+\log\left(\frac{m_\mu}{q_{\min}}\right)\,.
+\end{equation}
+Making the substitution $u^2 = 1/t -1$, $du^2 = -dt \, / t^2$ gives
+\begin{equation}
+\frac{d\sigma}{dx_+\,dt}=\frac{4\,Z^2\alpha^3}{m_\mu^2}\,
+\left[1-2\,x_+x_-+4\,x_+x_-t\,(1-t)\right]\,
+\log\left(\frac{m_\mu}{q_{\min}}\right) . \,
+\label{eq:sigmadxdt}
+\end{equation}
+Atomic screening and the finite nuclear radius may be taken into account by 
+multiplying the differential cross section determined by 
+Eq.\,(\ref{eq:MultiDiff2}) with the factor
+\begin{equation}\label{q5}
+  \left(F_a(q)-F_n(q)\,\right)^2\,,
+\end{equation}
+where $F_a$ and $F_n$ are atomic and nuclear form factors.
+Please note that after integrating Eq.~\ref{mupgen.b} over $\rho$, the 
+$q$-dependence is lost.
+
+\subsection{Procedure for the Generation of Muon - Anti-muon Pairs}
+
+Given the photon energy $E_\gamma$ and $Z$ and $A$ of the material in which 
+the $\gamma$ converts, the probability for the conversions to take place is 
+calculated according to the parametrized total cross section 
+Eq.\,(\ref{eq:sigpar}).  The next step, determining how the photon energy
+is shared between the $\mu^+$ and $\mu^-$, is done by generating $x_+$ 
+according to Eq.\,(\ref{eq:dSigxPlus}).  The directions of the muons are 
+then generated via the auxilliary variables $t,\,\rho,\,\psi$.  In more
+detail, the final state is generated by the following five steps, in which
+$R_{1,2,3,4,...}$ are random numbers with a flat distribution in the 
+interval [0,1]. The generation proceeds as follows.
+\\ \\
+{\bf 1)} Sampling of the positive muon energy $E_\mu^+ = x_+ \, E_\gamma$. \\
+This is done using the rejection technique.
+$x_+$ is first sampled from a flat distribution within kinematic limits 
+using
+\[ x_+ = x_{\rm min} + R_1 (x_{\rm max} - x_{\rm min}) \]
+and then brought to the shape of Eq.\,(\ref{eq:dSigxPlus}) by keeping all 
+$x_+$ which satisfy 
+\[ \left(1-\frac43\,x_+x_-\right)\frac{\log(W)}{\log(W_{\rm max})} < R_2 \,. \]
+Here $W_{\rm max}= W(x_+=1/2)$ is the maximum value of $W$, obtained for 
+symmetric pair production at $x_+=1/2$.  About 60\% of the events are kept 
+in this step.  Results of a Monte Carlo generation of $x_+$ are illustrated 
+in Fig.\,\ref{plot:xPlusGen}.  The shape of the histograms agrees with the 
+differential cross section illustrated in Fig.\,\ref{plot:dsigdx}.
+\begin{figure}[htpb]
+\center\includegraphics[scale=.7]{electromagnetic/standard/MuPgen/xPlusGen.eps}
+\caption{Histogram of generated $x_+$ distributions for beryllium at three 
+different photon energies.  The total number of entries at each energy is 
+$10^6$.}
+\label{plot:xPlusGen}
+\end{figure}
+\\ \\
+{\bf 2)} Generate $t ( = \frac{1}{\gamma^2 \theta^2 + 1} )$ . \\
+The distribution in $t$ is obtained from Eq.(\ref{eq:sigmadxdt}) as
+\begin{equation}\label{t}
+f_1(t)\,dt=\frac{1-2\,x_+x_-+4\,x_+x_-t\,(1-t)}
+{1+C_1/t^2}\,dt\,,\quad 0<t\le 1\,.
+\end{equation}
+with form factors taken into account by
+\begin{equation}\label{C_1}
+C_1=
+\frac{(0.35\,A^{0.27})^2}{x_+x_-\,E_\gamma/m_\mu }\,.
+\end{equation}
+In the interval considered, the function $f_1(t)$ will always be bounded 
+from above by
+\[
+\max [f_1(t)]=\frac{1-x_+x_-}{1+C_1}\;.\,
+\]
+For small $x_+$ and large $E_\gamma$, $f_1(t)$ approaches unity, as shown
+in Fig.\,\ref{f1t.eps}.
+\begin{figure}[htpb]
+\includegraphics[scale=.65]{electromagnetic/standard/MuPgen/f1t_10.eps}\hfill\includegraphics[scale=.65]{electromagnetic/standard/MuPgen/f1t_1000.eps}
+  \caption{The function $f_1(t)$ at $E_\gamma = 10\,{\rm GeV}$ (left) and
+  $E_\gamma = 1\,{\rm TeV}$ (right) in beryllium for different values of 
+  $x_+$.}
+\label{f1t.eps}
+\end{figure}
+
+\begin{figure}[htpb]
+% \begin{minipage}[t]{7.5cm}
+% \includegraphics[scale=.62]{electromagnetic/standard/MuPgen/f1tgen.eps}
+  \center\includegraphics[scale=.8]{electromagnetic/standard/MuPgen/f1tgen.eps}
+ \caption{Histograms of generated $t$ distributions for 
+  $E_\gamma = 10\,{\rm GeV}$ (solid line) and 
+  $E_\gamma = 100\,{\rm GeV}$ (dashed line) with $10^6$ events each.}
+ \label{f1t_gen.eps}
+% \end{minipage}
+\end{figure}
+% \hfill
+\begin{figure}[htpb]
+%  \begin{minipage}[t]{7.5cm}
+% { \center\includegraphics[scale=.6]{electromagnetic/standard/MuPgen/PsiGen.eps}
+  \center\includegraphics[scale=.8]{electromagnetic/standard/MuPgen/PsiGen.eps}
+  \caption{Histograms of generated $\psi$ distributions for beryllium at 
+  four different photon energies.}
+  \label{plot:PsiGen.eps}
+% }
+% \end{minipage}
+\end{figure}
+
+\noindent
+The Monte Carlo generation is done using the rejection technique.  About 
+70\% of the generated numbers are kept in this step.  Generated 
+$t$-distributions are shown in Fig.\,\ref{f1t_gen.eps}.
+\\ \\
+{\bf 3)} Generate $\psi$ by the rejection technique using $t$ generated in 
+the previous step for the frequency distribution
+\begin{equation}\label{q3}
+f_2(\psi) =\Big[1-2\,x_+x_-+4\,x_+x_-t\,(1-t)\,(1+\cos(2\psi))\Big]\;, \qquad 0\le\psi\le2\pi\,.
+\end{equation}
+The maximum of $f_2(\psi)$ is
+\begin{equation}
+\max [f_2(\psi)]=1-2\,x_+x_-\left[1-4\,t\,(1-t)\right]\,.\,
+\end{equation}
+Generated distributions in $\psi$ are shown in Fig.\,\ref{plot:PsiGen.eps}.
+\\ \\
+{\bf 4)} Generate $\rho$. \\
+%old The frequency distribution to generate is
+%old \begin{equation}
+%old f_2(\rho^2) = \frac{1}{\rho^2 + \kappa^2} \qquad  0 \leq \rho^2 \leq 1
+%old \end{equation}
+%old where $\kappa^2=1/W$ with the value of $W$ calculated in step 1. % according to Eq.\,\ref{eq:W}.
+%old The distribution can be generated by direct transformation.
+%old By integration
+%old \[
+%old F_2(x) = \int_0^x f_2(\rho^2) \, d \rho^2 = \log\left(1 + \frac{x}{\kappa^2} \right)
+%old \]
+%old with the inverse
+%old \[
+%old F_2^{-1}(x) =  \kappa^2 \, \left(e^x -1\right)\;.
+%old \]
+%old This is mapped by linear transform $a + b x$ to the range
+%old $(0,1)$ of the standard random generator.
+%old From
+%old \[ x=0 \quad : \qquad F_2^{-1}(a + b x) = \kappa^2 \, \left(e^a -1\right) = 0 \qquad  \Rightarrow \qquad a=0 \]
+%old and 
+%old \[ x=1 \quad : \qquad F_2^{-1}(a + b x) = \kappa^2 \, \left(e^b -1\right) = 1 \qquad  \Rightarrow \qquad 
+%old b=\log\left( 1+1/\kappa^2\right) = \log\left( 1+W^2\right) \; .\]
+%old The direct transformation $F_2^{-1}(a + b \, R)$, applied to the flat random distribution $R$ to generate $\rho^2$
+%old is therefore
+%old \begin{equation}
+%old \rho^2 = \kappa^2 \left(e^{R \log\left( 1+1/\kappa^2\right)} -1\right)
+%old = \kappa^2 \, \left[ \left( 1+1/\kappa^2\right)^R -1 \right]
+%old = \frac{\left( 1+W^2\right)^R -1 }{W^2} \;.
+%old \end{equation}
+%old Generated distributions in $\rho^2$ are shown in Fig.\,\ref{plot:rho2.eps}.
+%old The distribution is, with growing photon energy, increasingly peaked at 0.
+%old Note that $\rho$ does not depend on $t$. One could do step 3 before 2 or after step 4.
+The distribution in $\rho$ has the form
+\begin{equation}\label{rho1}
+f_3(\rho)\,d\rho=\frac{\rho^3\,d\rho}{\rho^4+C_2}\,,\quad
+0\le\rho\le \rho_{\rm max}\,,\,
+\end{equation}
+where
+\begin{equation}\label{rhomax}
+\rho_{\rm max}^2=\frac{1.9}{A^{0.27}}\,\left(\frac{1}{t}-1\right), \,
+\end{equation}
+and
+\begin{equation}\label{C2}
+C_2=\frac4{\sqrt{x_+x_-}}\left[\left(\frac{m_\mu}{2E_\gamma x_+x_-\,t}
+\right)^2+\left(\frac{m_e}{183 \, Z^{-1/3} \, m_\mu}\right)^2
+\right]^2\,.
+\end{equation}
+The $\rho$ distribution is obtained by a direct transformation applied to 
+uniform random numbers $R_i$ according to
+\begin{equation}\label{rho}
+\rho=\left[C_2(\exp(\beta\,R_i)-1)\right]^{1/4}\,,
+\end{equation}
+where
+\begin{equation}\label{beta}
+\beta=\log\left(\frac{C_2+\rho_{\rm max}^4}{C_2}\right)\,.
+\end{equation}
+Generated distributions of $\rho$ are shown in Fig.\,\ref{plot:rho.eps}
+
+\begin{figure}[htpb]
+% \begin{minipage}[t]{7.5cm}
+% \includegraphics[scale=.6]{electromagnetic/standard/MuPgen/rho.eps}
+\center\includegraphics[scale=.8]{electromagnetic/standard/MuPgen/rho.eps}
+\caption{Histograms of generated $\rho$ distributions for beryllium at 
+ two different photon energies.  The total number of entries at each energy
+ is $10^6$.}
+ \label{plot:rho.eps}
+% \end{minipage}
+\end{figure}
+% \hfill
+\begin{figure}[htpb]
+% \begin{minipage}[t]{7.5cm}
+% \includegraphics[scale=.6]{electromagnetic/standard/MuPgen/thetaPlus.eps}
+\center\includegraphics[scale=.8]{electromagnetic/standard/MuPgen/thetaPlus.eps}
+\caption{Histograms of generated $\theta_+$ distributions at different photon energies.}
+\label{plot:thetaPlus}
+% \end{minipage}
+\end{figure}
+
+\noindent
+{\bf 5)} Calculate $\theta_+,\theta_-$ and $\varphi$ from $t, \rho,\psi$ with
+\begin{equation}
+\gamma_\pm = \frac{E_\mu^\pm}{m_\mu} \qquad {\rm and} \qquad
+u=\sqrt{\frac1t-1}\,.
+\label{eq:gammau}
+\end{equation}
+according to
+\begin{equation}\label{s2}
+\theta_+=
+\frac{1}{\gamma_+}\,\left(u +\frac{\rho}{2}\,\cos\psi\right)\,,\quad \theta_-=
+\frac{1}{\gamma_-}\,\left(u -\frac{\rho}{2}\,\cos\psi\right)\, \quad {\rm and}
+\quad \varphi=\frac{\rho}{u} \, \sin\psi\, . \,
+\end{equation}
+The muon vectors can now be constructed from Eq.\,(\ref{eq:pvec}), where 
+$\varphi_0$ is chosen randomly between 0 and $2\pi$.
+Fig.\,\ref{plot:thetaPlus} shows distributions of $\theta_+$ at different 
+photon energies (in beryllium).  The spectra peak around $1/\gamma$ as 
+expected.
+
+The most probable values are $\theta_+\sim m_\mu/E_\mu^+ = 1 / \gamma_+$. In the small angle
+approximation used here, the values of $\theta_+$ and $\theta_-$
+can in principle be any positive value from 0 to $\infty$.
+In the simulation, this may lead (with a very small probability, of the 
+order of $m_\mu/E_\gamma$) to unphysical events in which $\theta_+$ or
+$\theta_-$ is greater than $\pi$.  To avoid this, a limiting angle 
+$\theta_{\rm cut}=\pi$ is introduced, and the angular sampling repeated, 
+whenever $\max(\theta_+,\,\theta_-)>\theta_{\rm cut}$ \, .
+
+\begin{figure}[htpb]
+\center\includegraphics[scale=.65]{electromagnetic/standard/MuPgen/Fig1.eps}
+\caption{Angular distribution of positive (or negative) muons.
+%old $E_\gamma=10\:{\rm GeV}$, $x_+=0.3$; iron.
+The solid curve represents
+the results of the exact calculations. The histogram is the simulated
+distribution. The angular distribution for pairs created in the field
+of the Coulomb centre (point-like target) is shown by the dashed curve
+for comparison.}
+\label{plot:Fig1}
+\end{figure}
+
+\begin{figure}[htpb]
+% \begin{minipage}[t]{7.5cm}
+% { \center\includegraphics[scale=.65]{electromagnetic/standard/MuPgen/Fig2.eps}
+\center\includegraphics[scale=0.8]{electromagnetic/standard/MuPgen/Fig2.eps}
+ \caption{Angular distribution in logarithmic scale.  The curve corresponds 
+  to the exact calculations and the histogram is the simulated 
+  distribution.}
+ \label{plot:Fig2}
+% }
+% \end{minipage}
+\end{figure}
+% \hfill
+\begin{figure}[htpb]
+% \begin{minipage}[t]{7.5cm}  
+% \includegraphics[scale=.65]{electromagnetic/standard/MuPgen/Fig3.eps}
+\center\includegraphics[scale=0.8]{electromagnetic/standard/MuPgen/Fig3.eps}
+\caption{Distribution of the difference of transverse momenta of positive 
+ and negative muons (with logarithmic x-scale).}
+  \label{plot:Fig3}
+% \end{minipage}
+\end{figure}
+
+Figs.\,\ref{plot:Fig1},\ref{plot:Fig2} and \ref{plot:Fig3} show
+distributions of the simulated angular characteristics of muon pairs in 
+comparison with results of exact calculations.  The latter were obtained 
+by means of numerical integration of the squared matrix elements with 
+respective nuclear and atomic form factors.  All these calculations were 
+made for iron, with $E_\gamma=10\,{\rm GeV}$ and $x_+=0.3$.  As seen from 
+Fig.\,\ref{plot:Fig1}, wide angle pairs (at low values of the argument in 
+the figure) are suppressed in comparison with the Coulomb center 
+approximation.  This is due to the influence of the finite nuclear size
+which is comparable to the inverse mass of the muon.  Typical angles of 
+particle emission are of the order of
+$1/\gamma_\pm=m_\mu/E_\mu^\pm$ (Fig.\,\ref{plot:Fig2}). Fig.\,\ref{plot:Fig3}
+illustrates the influence of the momentum transferred to the target on the 
+angular characteristics of the produced pair.  In the frame of the often 
+used model which neglects target recoil, the pair particles would be 
+symmetric in transverse momenta, and coplanar with the initial photon.
+
+
+\subsection{Status of this document}
+28.05.02 created by H.Burkhardt. \\
+01.12.02 re-worded by D.H. Wright \\
+
+\begin{latexonly}
+
+\begin{thebibliography}{99}
+
+\bibitem{MuPgen}
+H.~Burkhardt, S.~Kelner, and R.~Kokoulin, ``Monte Carlo Generator for Muon 
+Pair Production''. CERN-SL-2002-016 (AP) and CLIC Note 511, May 2002.
+
+\bibitem{Kelner:1995hu}
+S.~R. Kelner, R.~P. Kokoulin, and A.~A. Petrukhin, ``About cross section 
+for high energy muon bremsstrahlung,''. Moscow Phys. Eng. Inst. 024-95, 1995.
+  31pp.
+
+\end{thebibliography}
+
+\end{latexonly}
+
+\begin{htmlonly}
+
+\subsection{Bibliography}
+
+\begin{enumerate}
+\item H.~Burkhardt, S.~Kelner, and R.~Kokoulin, 
+``Monte Carlo Generator for Muon 
+Pair Production''. CERN-SL-2002-016 (AP) and CLIC Note 511, May 2002.
+
+\item S.~R. Kelner, R.~P. Kokoulin, and A.~A. Petrukhin, ``About cross section 
+for high energy muon bremsstrahlung,''. Moscow Phys. Eng. Inst. 024-95, 1995.
+  31pp.
+\end{enumerate}
+
+\end{htmlonly}
+
+
Index: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/electromagnetic/standard/hion.tex
===================================================================
--- trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/electromagnetic/standard/hion.tex	(revision 1211)
+++ trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/electromagnetic/standard/hion.tex	(revision 1222)
@@ -244,14 +244,9 @@
 \subsubsection{Nuclear Stopping} 
 
-For scaled energies below $T_{lim} = 2 MeV$ the contribution of non-ionizing energy loss
-needs to be taken into account. The additional energy loss due to {\it nuclear stopping power}
-$\Delta T_N \Delta s$ is added the the energy loss. The process {\it G4ionIonisation}
-has a flag, which allows to switch on or off this correction. For that the method\\
-\\
-{\it G4ionIonisation::ActivateNuclearStopping(G4bool)} 
-\\
-\\
-can be used. By default 
-this correction is active and the ICRU'49 parameterisation \cite{hion.ICRU49} is used.
+Nuclear stopping due to elastic ion-ion scattering since Geant4 v9.3
+can be simulated with the continuous process
+{\it G4NuclearStopping}. By default 
+this correction is active and the ICRU'49 parameterisation \cite{hion.ICRU49} is used, 
+which is implemented in the model class {\it G4ICRU49NuclearStoppingModel}.  
 
 
Index: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/electromagnetic/standard/msc.tex
===================================================================
--- trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/electromagnetic/standard/msc.tex	(revision 1211)
+++ trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/electromagnetic/standard/msc.tex	(revision 1222)
@@ -15,6 +15,4 @@
 \item
 G4MuMultipleScattering.
-\item
-G4MultipleScattering;
 \end{itemize}
 For concrete simulation the {\it G4VMscModel} interface is used, which is an extension
@@ -22,15 +20,16 @@
 \begin{itemize}
 \item
-G4UrbanMscModel;
+G4UrbanMscModel90 - Geant4 v9.0 applied to muons, hadrons and ions;
 \item
-G4UrbanMscModel2;
+G4UrbanMscModel92 - Geant4 v9.2 (current default) applied for electron and positron;
 \item
-G4UrbanMscModel90;
+G4UrbanMscModel93 - Geant4 v9.3 applied for electron and positron for Option2, Option3 and other EM Physics Lists;
 \item
-G4WentzelVIModel.
+G4GoudsmitSaundersonModel - for electrons and positrons (beta-version);
+\item
+G4WentzelVIModel - for muons and hadrons, should be included in Physics Lists together with G4CoulombScattering process;
 \end{itemize}
-The last model is not yet in the production mode, so below we will concentrate 
-on models developed by L.~Urban \cite{msc.urban}. Today default model for 
-electrons and positrons is {\it G4UrbanMscModel2}, for hadrons - {\it G4UrbanMscModel90}.
+The last models are not yet in the production mode, so below we will concentrate 
+on models developed by L.~Urban \cite{msc.urban}. 
 
 \subsection{Introduction}
@@ -811,4 +810,5 @@
  08.12.08  revised by L. Urb\'an, for Geant4 V9.2 \\
  11.12.08  minor revision by V. Ivanchenko \\
+ 11.12.09  minor revision by V. Ivanchenko, for Geant4 v9.3 \\
  
 \begin{latexonly}
