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diff --git a/doc/IGE335/Section5.01.tex b/doc/IGE335/Section5.01.tex new file mode 100644 index 0000000..ba4f706 --- /dev/null +++ b/doc/IGE335/Section5.01.tex @@ -0,0 +1,50 @@ +\subsection{Scattering cross sections}\label{sect:ExXSData} + +In DRAGON, the angular dependence of the +scattering cross section is expressed in a Legendre series expansion of +the form: + $$ +\Sigma_{s}(\Omega\cdot\Omega')=\Sigma_{s}(\mu)= +\sum_{l=0}^{L}\left({{(2l+1)}\over{4\pi}}\right)\Sigma_{s,l}P_{l}(\mu). + $$ +Since the Legendre polynomials satisfy the following +orthogonality conditions: + $$ +\int_{-1}^{1} d\mu P_{l}(\mu)P_{m}(\mu) = +\left({{2\delta_{l,m}}\over{(2l+1)}}\right), + $$ +we will have + $$ +\Sigma_{s,l}=\int_{-1}^{1}d\mu\int_{0}^{2\pi}d\varphi\Sigma_{s}(\mu)P_{l}(\mu)= +2\pi \int_{-1}^{1}d\mu\Sigma_{s}(\mu)P_{l}(\mu). + $$ + +Let us now consider the following three-group (\dusa{ngroup}=3) isotropic and +linearly anisotropic scattering cross sections ($L$=\dusa{naniso}=2) given by: + +\begin{center} +\begin{tabular}{|llccc|}\hline\hline +$l$ & $g$ & $\Sigma_{s,l}^{g\to 1}$ (\xsunit) + & $\Sigma_{s,l}^{g\to 2}$ (\xsunit) + & $\Sigma_{s,l}^{g\to 3}$ (\xsunit) \\ \hline + & 1 & 0.90 & 0.80 & 0.00 \\ +0 & 2 & 0.00 & 0.70 & 0.60 \\ + & 3 & 0.00 & 0.30 & 0.40 \\ \hline + & 1 & 0.09 & 0.05 & 0.08 \\ +1 & 2 & 0.00 & 0.07 & 0.06\\ + & 3 & 0.03 & 0.00 & 0.04 \\ \hline\hline +\end{tabular} +\end{center} + +\noindent +In DRAGON this scattering cross section must be entered as + +\begin{verbatim} +SCAT (* L=0 *) 1 1 (* 3->1 *) (* 2->1 *) (* 1->1 *) 0.90 + 3 3 (* 3->2 *) 0.30 (* 2->2 *) 0.70 (* 1->2 *) 0.80 + 2 3 (* 3->3 *) 0.40 (* 2->3 *) 0.60 (* 1->3 *) +SCAT (* L=1 *) 3 3 (* 3->1 *) 0.03 (* 2->1 *) 0.00 (* 1->1 *) 0.09 + 2 2 (* 3->2 *) (* 2->2 *) 0.07 (* 1->2 *) 0.05 + 3 3 (* 3->3 *) 0.04 (* 2->3 *) 0.06 (* 1->3 *) 0.08 +\end{verbatim} + |
