scattering-theory.tex

\end{equation}

in which $d$ has been used to represent $|E_\lambda-E-i\Gamma/2|^2=(E-E_\lambda)^2+(\Gamma/2)^2$. Similarly, the elastic cross section, Eq. (II B1 a.3)\ref{eq:elastic-in-terms-of-X}, can be expressed as

\begin{equation}
    \begin{aligned}
        \sigma_{\text {elastic }}(E)=& \frac{4 \pi}{k_{\alpha}^{2}} \sum_{J} g_{J \alpha} \sum_{c}\left[\sin ^{2} \phi_{c}\left(1-2 \frac{\Gamma \Gamma_{1}}{4 d}\right)\right.\\
        &\left.-\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{2 d} \sin \left(2 \phi_{c}\right)+\left(\frac{\Gamma \Gamma_{1}}{4 d}\right)^{2}+\left(\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{2 d}\right)^{2}\right]
    \end{aligned}
\end{equation}

\noindent
which reduces to

\begin{equation}
    \begin{aligned}
        \sigma_{\text {elastic }}(E)=\frac{2 \pi}{k_{\alpha}^{2}} \sum_{J} g_{J \alpha} \sum_{c} & {\left[1-\cos 2 \phi_{c}\left(1-\frac{\Gamma \Gamma_{1}}{2 d}\right)\right.} \\
        &\left.-\sin 2 \phi_{c} \frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{d}-\frac{\Gamma_{1}\left(\Gamma_{\gamma}+\Gamma_{2}\right)}{2 d}\right].
    \end{aligned}
\end{equation}

\noindent
The reaction cross section, Eq. (II B1 a.4)\ref{eq:reax-in-terms-of-X}, becomes

\begin{equation}
    \begin{aligned}
        \sigma_{\text {reaction }}(E) &=\frac{4 \pi}{k_{\alpha}^{2}} g\left[\left(\frac{\Gamma \sqrt{\Gamma_{1} \Gamma_{2}}}{4 d}\right)^{2}+\left(\frac{\left(E-E_{\lambda}\right) \sqrt{\Gamma_{1} \Gamma_{2}}}{2 d}\right)^{2}\right] \\
        &=\frac{\pi g}{k_{\alpha}^{2}}\left[\frac{\Gamma_{1} \Gamma_{2}}{d}\right],
    \end{aligned}
\end{equation}

\noindent
and, finally, the capture cross section, Eq. (II B1 a.6)\ref{eq:sig-cap-in-terms-of-X}, is

\begin{equation}
    \begin{aligned}
        \sigma_{\text {capture }}(E) &=\frac{4 \pi g}{k_{\alpha}^{2}}\left[\frac{\Gamma \Gamma_{1}}{4 d}-\left\{\left(\frac{\Gamma \Gamma_{1}}{4 d}\right)^{2}+\left(\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{2 d}\right)^{2}\right.\right.\\
        &\qquad\left.\left.+\left(\frac{\Gamma \sqrt{\Gamma_{1} \Gamma_{2}}}{4 d}\right)^{2}+\left(\frac{\left(E-E_{\lambda}\right) \sqrt{\Gamma_{1} \Gamma_{2}}}{2 d}\right)^{2}\right\}\right] \\
        &= \frac{4 \pi g}{k_{\alpha}^{2}}\left[\frac{\Gamma \Gamma_{1}}{4 d}-\left\{\frac{\Gamma_{1}^{2}}{4 d}+\frac{\Gamma_{1} \Gamma_{2}}{4 d}\right\}\right]=\frac{\pi g}{k_{\alpha}^{2}}\left[\frac{\Gamma_{1} \bar{\Gamma}_{\gamma}}{d}\right] .
    \end{aligned}
\end{equation}

\newpage
\noindent
\textbf{Angular distributions}

Angular distributions (elastic, inelastic, or other reaction) cross sections for incident neutrons can be calculated from Reich-Moore resonance parameters. Following Blatt and Biedenharn [JB52]\cite{blatt-biedenharn-1952} with some notational changes, the angular distribution cross section in the center-of-mass system may be written

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\subsection{Simulation of Full R-Matrix} \label{subsec:r-matrix-full}
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\subsection{Breit-Wigner Approximations} \label{subsec:r-matrix-BW}
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\subsection{Direct Capture Component} \label{subsec:direct-capture}
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\section{Details and Conventions Used In Sammy} \label{sec:details-and-conventions}
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\textbf{Note: Subject to change with inclusion of AMPX R-matrix engine!!!}


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\subsection{Spin and Angular Momentum Conventions} \label{subsec:spin-conventions}
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\subsection{Kinematics} \label{subsec:kinematic-conventions}
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\subsection{Evaluation of Hard-Sphere Phase Shift} \label{subsec:phase-shift-conventions}
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\subsection{Modifications for Charged Particles} \label{subsec:charged-particle-conventions}
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\subsection{Inverse Reactions (Reciprocity)} \label{subsec:inverse-reactions-conventions}
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\section{Derivatives} \label{sec:derivatives}
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In order to make use of sophisticated fitting procedures such as Bayes' equations (Section IV \ref{} of this manual), it is necessary to know the partial derivatives of the theory with respect to the parameters to be fitted (the ``varied parameters''). The easiest method for calculating derivatives of cross sections with respect to resonance parameters is to use a numerical difference approximation, of the form

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\subsection{Derivatives for Reich-Moore Approximation} \label{subsec:derivs-RM}
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\subsection{Derivatives for MLBW and SLBW Approximations} \label{subsec:derivs-BW}
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\subsection{Details Involving Derivatives} \label{subsec:derivs-details}
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