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\section[Synchrotron Radiation]{Synchrotron Radiation}

Synchrotron radiation photons are produced when ultra-relativistic electrons
travel along an approximately circular path.  In the following treatment, 
the magnetic field is assumed to be constant and uniform, and the radius of
curvature of the electron is assumed to be constant over its trajectory.

\subsection{Spectral and Angular Distributions of Synchrotron Radiation}

The spectral distribution of the mean number of synchrotron radiation
photons, $d\bar{N}/d\omega$, produced by an ultra-relativistic electron 
along a circular trajectory of length $L$, can be expressed in terms of the 
mean energy loss spectrum $d\bar{\Delta}/d\omega$ \cite{synch.maier}:

\begin{equation}
\label{synch.a}
\frac{d\bar{N}}{d\omega} = \frac{1}{\omega}\frac{d\bar{\Delta}}{d\omega} = 
\frac{\sqrt{3}}{2\pi}\alpha\left(\frac{L\gamma}{R}\right)\frac{1}{\omega_{c}}
\int_{\omega/\omega_{c}}^{\infty}K_{5/3}(\eta)d\eta   .
\end{equation}

Here,
\begin{eqnarray*}
\omega   &  & \mbox{photon energy} \\
\alpha   &  & \mbox{fine structure constant} \\
R        &  & \mbox{instantaneous radius of curvature of the trajectory} \\ 
K        &  & \mbox{Macdonald function} \\
\omega_c = 1.5\beta(\hbar c/R)\gamma^3 &  & \mbox{characteristic energy of synchrotron radiation.} \\   
\end{eqnarray*} 
$\beta$ is the ratio of the electron velocity $v$ to $c$, 
$\gamma = 1/\sqrt{1 - \beta^2}$, and $\eta$ is an arbitrary integration
variable.  In the SI system of units: $R(m) = P(GeV/c)/0.3B_{\bot}(T)$ , 
where $B_{\bot}$ is the component of magnetic flux density perpendicular to 
the electron velocity, and $P$ is the electron momentum.

In order to simulate the energy spectrum of synchrotron radiation using
the Monte Carlo method, $\bar{N}_{>\omega}$, the mean number of photons
above a given energy $\omega$, must be determined.  This is done by
integrating Eq.~\ref{synch.a} over energy, after first transforming  
$d\bar{N}/d\omega$ by using the integral representation of the Macdonald 
function \cite{synch.abram}:

\begin{eqnarray}
\label{synch.b}
\bar{N}_{>\omega}& = &
\int_{\omega}^{\infty}\frac{d\bar{N}}{d\omega'}d\omega' \nonumber\\
& = &
\frac{\sqrt{3}}{2\pi}\alpha\left(\frac{L\gamma}{R}\right)
\int_0^{\infty}\frac{\cosh\left(\frac{5}{3}t\right)}{\cosh^2(t)}
\exp\left[-\frac{\omega}{\omega_c}\cosh(t)\right]dt .
\end{eqnarray}
Here, $t$ is also an arbitrary integration variable.  The latter integral 
is calculated numerically by the quadrature Laguerre 
formula \cite{synch.korn}.  Calculations indicate that about 50 roots of 
the Laguerre polynomials are required in order for the accuracy of the 
integral estimation to be better than $10^{-4}$ \cite{synch.bag}.  

The Monte Carlo method also requires the mean number of synchrotron 
radiation photons at all energies, $\bar{N}$  (= $\bar{N}_{>0}$), in order 
to determine the next occurrence of synchrotron radiation along a 
trajectory, and to normalize the spectral distribution of the radiation.
Setting $\omega = 0$ in Eq.~\ref{synch.b} yields
\begin{eqnarray}
\bar{N} = \bar{N}_{>0}& = & 
\frac{\sqrt{3}}{2\pi}\alpha\left(\frac{L\gamma}{R}\right)
\int_0^{\infty}\frac{\cosh\left(\frac{5}{3}t\right)}{\cosh^2(t)}dt \nonumber\\
& = &
\frac{5}{2\sqrt{3}}\alpha\left(\frac{L\gamma}{R}\right) \approx 
10^{-2}\left(\frac{L\gamma}{R}\right) .
\end{eqnarray}

Qualitatively this result can be manipulated using the fact that the mean 
number of photons produced along the formation zone length 
$z \approx R/\gamma$ is proportional to $\alpha$.  Then for length $L$, 
$\bar{N} \approx \alpha L/(R/\gamma)$.  Note that when $\gamma\gg1$, and
$R \sim \gamma$, $\bar{N}$ does not depend on the electron energy but is 
defined by the values of $L$ and $B_{\bot}$ only.  Instead, it is the mean 
energy loss due to synchrotron radiation $\bar\Delta$, corresponding to a 
trajectory of length $L$, that displays the characteristic 
relativistic rise:

\begin{equation}
\bar{\Delta} = \int_0^{\infty}\omega\frac{d\bar{N}}{d\omega}d\omega =
\frac{2}{3}\alpha\hbar c\left(\frac{L\gamma^2}{R^2}\right)\beta\gamma^2 =   
\frac{8\bar{N}}{15\sqrt{3}}\omega_{c} 
\approx 0.31\bar{N}\omega_{c} \sim \gamma^2 .
\end{equation}

The angular distribution of synchrotron radiation produced by 
ultra-relativistic electrons shows a clear 'searchlight' effect.  Most of
the photons are radiated within an angular range of order $1/\gamma$
centered on the electron trajectory direction.  In the interesting region 
of $\gamma > 10^3$ the angular resolution of X-ray and gamma detectors 
usually does not allow the details of the angular distribution to be 
measured.  Therefore, the angular distribution is set to be flat in the 
range $0 - 1/\gamma$.

\subsection{Simulating Synchrotron Radiation}

The distance $x$ along the electron/positron trajectory to the next 
occurrence of a synchrotron radiation photon is simulated according to the
exponential distribution, $exp(-x\bar{N}/L)$.  The energy $\omega$ of the 
photon is simulated according to the distribution 
$\bar{N}_{>\omega}/\bar{N}$.  The direction of the photon 
($\theta$, $\varphi$) is generated relative to the local $z$-axis which is 
taken to be along the instantaneous direction of the electron.  $\theta$
and $\varphi$ are distributed randomly in the ranges [0, $1/\gamma$] and 
[0, $2\pi$], respectively.

\subsection{Status of this document}
15.10.98 created by V.Grichine \\
02.12.02 re-written by D.H. Wright \\

\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{synch.maier}  R. Maier,
{\em Synchrotron Radiation, CERN Report 91-04, 97-115} (1991).

\bibitem{synch.abram} Edited by M. Abramovwitz and I.A. Stegan,
{\em Handbook of Mathemaatical Functions,
 NBS Applied Mathematics Series 55} (1964).

\bibitem{synch.korn}G.A. Korn and T.M. Korn,
{\em Mathematical Handbook for scientists and 
    engineers, McGRAW-HILL BOOK COMPANY, INC} (1961).

\bibitem{synch.bag} A.V. Bagulya and V.M. Grichine,
{Bulletin of Lebedev Institute, no.9-10, 7 } (1998).

\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item  R. Maier,
{\em Synchrotron Radiation, CERN Report 91-04, 97-115} (1991).

\item Edited by M. Abramovwitz and I.A. Stegan,
{\em Handbook of Mathemaatical Functions,
 NBS Applied Mathematics Series 55} (1964).

\item G.A. Korn and T.M. Korn,
{\em Mathematical Handbook for scientists and 
    engineers, McGRAW-HILL BOOK COMPANY, INC} (1961).

\item A.V. Bagulya and V.M. Grichine,
{Bulletin of Lebedev Institute, no.9-10, 7 } (1998).

\end{enumerate}

\end{htmlonly}