scattering-theory.tex

    \end{aligned}
\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

\begin{equation}
    \frac{d\sigma_{\alpha\alpha'}}{d\Omega_{CM}} = \sum_L B_{L\alpha\alpha'}(E)P_L(\cos\beta),
\end{equation}

\noindent
in which the subscript $\alpha\alpha'$ indicates which type of cross section is being considered (i.e., $\alpha$ represents the entrance particle pair and $\alpha'$ represents the exit pair). $P_L$ is the Legendre polynomial of degree $L$, and $\beta$ is the angle of the outgoing neutron (or other particle) relative to the incoming neutron in the center-of-mass system. The coefficients $B_{L\alpha\alpha'}(E)$ are given by

\begin{equation}\label{eq:ang-B-coeff}
    \begin{aligned}
        B_{L\alpha\alpha'}(E) &= \frac{1}{4k_\alpha}^2\sum_{J_1}\sum_{J_2}\sum_{l_1s_1}\sum_{l_1's_1'}\sum_{l_2s_2}\sum_{l_2's_2'}\frac{1}{(2i+1)(2I+1)} \\
        &\qquad\times G_{\{l_1s_1l_1's_1'J_1\}\{l_2s_2l_2's_2'J_2\}L} \text{Re}\left[(\delta_{c_1c_1'}-U_{c_1c_1'})(\delta_{c_2c_2'}-U^*_{c_2c_2'})\right]
    \end{aligned}
\end{equation}

\noindent
in which the various summations are to be interpreted as follows:
\begin{enumerate}
    \item sum over all spin groups defined by spin $J_1$ and the implicit associated parity
    \item sum over all spin groups defined by spin $J_2$ and the implicit associated parity
    \item sum over the entrance channels $c_1$ belonging to the $J_1$ spin group and having particle pair $\alpha$, with orbital angular momentum $l_1$ and channel spin $s_1$ [i.e.,$c_1=(\alpha,l_1,s_1,J_1)$]
    \item sum over the exit channels $c_1'$ in $J_1$ spin group with particle-pair $\alpha'$ , orbital angular momentum $l_1'$ , and channel spin $s_1'$ [i.e., $c_1'=(\alpha',l_1',s_1',J_1)$]
    \item sum over entrance channels $c_2$ in $J_2$ spin group where $c_2=(\alpha,l_2,s_2,J_2)$
    \item sum over exit channels $c_2'$ in $J_2$ spin group where $c_2'=(\alpha',l_2',s_2',J_2)$
\end{enumerate}

\noindent
Also note that $i$ and $I$ are the spins of the two particles (projectile and target nucleus) in particle-pair $\alpha$.

The geometric factor $G$ can be exactly evaluated as a product of terms

\begin{equation}
    G_{\{l_1s_1l_1's_1'J_1\}\{l_2s_2l_2's_2'J_2\}L} = A_{l_1s_1l_1's_1';J_1}A_{l_2s_2l_2's_2';J_2}D_{l_1s_1l_1's_1'l_2s_2l_2's_2';LJ_1J_2},
\end{equation}

\noindent
where the factor $A_{l_1s_1l_1's_1';J_1}$ is of the form

\begin{equation}\label{eq:ang-A-term}
    A_{l_1s_1l_1's_1';J_1} = \sqrt{(2l_1+1)(2l_1'+1)}(2J_1+1)\Delta(l_1J_1s_1)\Delta(l_1'J_1s_1').
\end{equation}

\noindent
The expression for $D$ is

\begin{equation}\label{eq:ang-D-term}
    \begin{aligned}
        &D_{l_{1} s_{1} l_{1} s_{1} l_{2} s_{2} l_{2}^{\prime} s_{2}^{\prime} ; L J_{1} J_{2}}=(2 L+1) \Delta^{2}\left(J_{1} J_{2} L\right) \Delta^{2}\left(l_{1} l_{2} L\right) \Delta^{2}\left(l_{1}^{\prime} l_{2}^{\prime} L\right) \\
        &\times w\left(l_{1} J_{1} l_{2} J_{2}, s_{1} L\right) w\left(l_{1}^{\prime} J_{1} l_{2}^{\prime} J_{2}, s_{1}^{\prime} L\right) \delta_{s_{1} s_{2}} \delta_{s_{1}^{\prime} s_{2}^{\prime}}(-1)^{s_{1}-s_{1}^{\prime}} \\
        &\times \frac{n !(-1)^{n}}{\left(n-l_{1}\right) !\left(n-l_{2}\right) !(n-L) !} \frac{n^{\prime} !(-1)^{n^{\prime}}}{\left(n^{\prime}-l_{1}^{\prime}\right) !\left(n^{\prime}-l_{2}^{\prime}\right) !\left(n^{\prime}-L\right) !} \quad,
    \end{aligned}
\end{equation}

\noindent
in which $n$ is defined by

\begin{equation}\label{eq:2n-for-D}
    2n=l_1+l_2+L
\end{equation}

\noindent
$D$ is zero if $l_1+l_2+L$ is an odd number. A similar expression defines $n'$. The $\Delta^2$ term is given by

\begin{equation}
    \Delta^2(abc)= \frac{(a+b-c)!(a-b+c)!(-a+b+c)!}{(a+b+c+1)!} \;,
\end{equation}

\noindent
for which the arguments $a$, $b$, and $c$ are to be replaced by the appropriate values given in Eqs. (II B1 b.4)\ref{eq:ang-A-term} and (II B1 b.5)\ref{eq:ang-D-term}. The expression for $\Delta^2(abc)$ implicitly includes a selection rule for the arguments; that is, the quantized vector sum must hold,

\begin{equation}
    \vec{a}+\vec{b}=\vec{c} \qquad or \qquad |a-b|\leq c\leq a+b
\end{equation}

\noindent
with $c$ being either integer or half-integer. The quantity $w$ in Eq. (II B1 b.5)\ref{eq:ang-D-term} is defined as

\begin{equation}
    \begin{aligned}
        w\left(l_{1} J_{1} l_{2} J_{2}, s L\right)&=\sum_{k=k \min }^{k \max } \frac{(-1)^{k+l_{1}+J_{1}+l_{2}+J_{2}}(k+1) !}{\left(k-\left(l_{1}+J_{1}+s\right)\right) !\left(k-\left(l_{2}+J_{2}+s\right)\right) !} \\
        &\qquad \times \frac{1}{\left(k-\left(l_{1}+l_{2}+L\right)\right) !\left(k-\left(J_{1}+J_{2}+L\right)\right) !} \\
        &\qquad \times \frac{1}{\left(l_{1}+J_{1}+l_{2}+J_{2}-k\right) !\left(l_{1}+J_{2}+s+L-k\right) !\left(l_{2}+J_{1}+s+L-k\right) !}
    \end{aligned}
\end{equation}

\noindent
(and similarly for the primed expression), where $kmin$ and $kmax$ are chosen such that none of the arguments of the factorials are negative. That is,

\begin{equation}
    \begin{aligned}
        kmin &= \text{max}\left\{(l_1+J_1+s),(l_2+J_2+s),(l_1+l_2+L),(J_1+J_2+L)\right\} \\
        kmax &= \text{min}\left\{(l_1+J_1+l_2+J_2),(l_1+J_2+s+L),(l_2+J_1+s+L)\right\}.
    \end{aligned}
\end{equation}

\newpage
\noindent
\textbf{Angular distributions: Single-channel case}

\noindent
For some situations, these equations can be greatly simplified. When the target spin is zero and there are no possible reactions (no fission, no inelastic, no other reactions), then each spin group will consist of a single channel (the elastic channel). In this case, the coefficients $B_{L\alpha\alpha'}(E)$ reduce to

\begin{equation}
    \begin{aligned}
        B_{L \alpha \alpha}(E)=\frac{1}{4 k_{\alpha}^{2}} \sum_{J_{1}} & \sum_{J_{2}} \sum_{c_{1}=\left(\alpha l_{1} s_{1} J_{1}\right)} \sum_{c_{2}=\left(\alpha l_{2} s_{2} J_{2}\right)} G_{\left\{l_{1} s_{1} l_{1} s_{1} J_{1}\right\}\left\{l_{2} s_{2} l_{2} s_{2} J_{2}\right\} L} \operatorname{Re}\left[\left(1-U_{c_{1} c_{1}}\right)\left(1-U_{c_{2} c_{2}}^{*}\right)\right] \\
        & \times \frac{1}{\left(2 i_{a}+1\right)\left(2 i_{b}+1\right)}
    \end{aligned}
\end{equation}

\noindent
where the existence of only one channel requires that the primed quantities of Eq.(II B1 b.2)\ref{eq:ang-B-coeff} be equal to the unprimed (e.g., $\alpha=\alpha'$). The geometric factor $G$ becomes

\begin{equation}
    G_{\left\{l_{1} s_{1} l_{1} s_{1} J_{1}\right\}\left\{l_{2} s_{2} l_{2} s_{2} J_{2}\right\} L}=A_{l_{1} s_{1} l_{1} s_{1} ; J_{1}} A_{l_{2} s_{2} l_{2} s_{2} ; J_{2}} D_{l_{1} s_{1} l_{1} s_{1} l_{2} s_{2} l_{2} s_{2} ; L J_{1} J_{2}}
\end{equation}

\noindent
in which the factor $A$ reduces to the simple form

\begin{equation}
    A_{l_1s_1l_1s_1;J_1} = (2l_1+1)(2J_1+1)\Delta^2(l_1J_1s_1),
\end{equation}

\noindent
and the expression for $D$ reduces to

\begin{equation}
    \begin{aligned}
        D_{l_{1} s_{1} l_{1} s_{1} l_{2} s_{2} l_{2} s_{2} ; L J_{1} J_{2}} &=(2 L+1) \Delta^{2}\left(J_{1} J_{2} L\right) \Delta^{4}\left(l_{1} l_{2} L\right) \\
        &\qquad\times w^{2}\left(l_{1} J_{1} l_{2} J_{2}, s_{1} L\right) \delta_{s_{1} s_{2}}\left[\frac{n !}{\left(n-l_{1}\right) !\left(n-l_{2}\right) !(n-L) !}\right]^{2},
    \end{aligned}
\end{equation}

\noindent
in which n is again defined as in Eq. (II B1 b.6)\ref{eq:2n-for-D}.

\newpage
\noindent
\textbf{Specifying individual reaction types}

Early versions of SAMMY permitted users to specify ``inelastic'', ``fission'', and ``reaction'' data. However, the tacit assumption was that all the exit channels are relevant to the type of data being used. If, for example, three exit channels were specified as (1) inelastic, (2) first fission channel, and (3) second fission channel, then any calculation for ``inelastic'', ``fission'', or ``reaction'' data types would automatically include all three exit channels in the final state.

Hence, in early versions of SAMMY, true inelastic cross sections (for example) would be calculated only if all of the following conditions were met:

\begin{enumerate}
    \item Either ``inelastic'', ``fission'', or ``reaction'' was specified as the data type in the INPut file, card set 8.
    \item The exit channel description was appropriate for inelastic channels: The INPut file noted that penetrabilities were to be calculated (LPENT = 1 on line 2 of card set 10.1) and also provided a non-zero value for the excitation energy.
    \item No fission channel (or other exit channel) was defined in the INPut file (and PARameter file).
    \item[Note:] Beginning with release M5 of the SAMMY code, it is now possible to include only a subset of the exit channels in the outgoing final state. The third condition in the list above is no longer necessary, but is replaced by another (less restrictive) condition:
    \item[3.] Exit channels that are not inelastic have a flag (``1'' in column 18 of line 2 of card set 10.1 or card set 10.2 of the INPut file), denoting that this channel does not contribute to the final state.
\end{enumerate}

\noindent
(Similar considerations hold, of course, for any other reaction type, not only for inelastic.)

With release 7.0.0 of the SAMMY code in 2006, a more intuitive input is possible. When channels are specified using either of the particle-pair options (see card set 4 or 4a of Table VIA.1), then the data type line (card set 8 of Table VIA.1) may be used to specify the name(s) of the particle pair(s) to be included in the final-state reaction. Specifically, beginning in the first column of card set 8, include the phrase

\texttt{FINAL-state particle pairs are}

\noindent
or

\texttt{PAIRS in final state =}

\noindent
(Only the first five characters are required, the others are optional.) Elsewhere on the same line, give the eight-character designation of the particle pair(s) to be included in the final-state reaction. Only channels involving those particle pairs will be included in the final state; any channels not involving those particle pairs will not be included. (Caution: The particle pair name must be exactly as it appears in the INPut file, including capitalization.)

The same two command lines may be used for angular distributions with specific final states, provided the phrase ``ANGULar distribution'' is given later on the same line.

See test case tr159 for an example which includes three reactions, one being (n,$\alpha$) and the other two inelastic (n,n'). Various options for input are given in this test case.

Run ``k'' of test case tr112 shows an example for the angular distribution of a reaction cross
section.

\newpage
\noindent
\textbf{External R-function}

When generating cross sections via R-matrix theory, it is important to include contributions from all resonances, even those outside the energy range of the data. Tails from negative-energy resonances (which may correspond to bound states) and from higher-lying resonances can contribute significantly to the ``background'' of the R-matrix and must therefore not be omitted. There are infinitely many of these resonances, so approximations must be made.

The usual approximation is to use pseudo or dummy resonances to approximate the effect of the infinite number of outlying resonances. The energy associated with a dummy resonance must be outside the energy region for which the analysis is valid.

For discussion regarding two different philosophies for determining appropriate choices of dummy resonances, see Leal et al. [LL99] and Fröhner and Bouland [FF01].

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simulation of Full R-Matrix} \label{subsec:r-matrix-full}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Breit-Wigner Approximations} \label{subsec:r-matrix-BW}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Direct Capture Component} \label{subsec:direct-capture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Details and Conventions Used In Sammy} \label{sec:details-and-conventions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textbf{Note: Subject to change with inclusion of AMPX R-matrix engine!!!}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spin and Angular Momentum Conventions} \label{subsec:spin-conventions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Kinematics} \label{subsec:kinematic-conventions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Evaluation of Hard-Sphere Phase Shift} \label{subsec:phase-shift-conventions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Modifications for Charged Particles} \label{subsec:charged-particle-conventions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Inverse Reactions (Reciprocity)} \label{subsec:inverse-reactions-conventions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Derivatives} \label{sec:derivatives}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Derivatives for Reich-Moore Approximation} \label{subsec:derivs-RM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Derivatives for MLBW and SLBW Approximations} \label{subsec:derivs-BW}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Details Involving Derivatives} \label{subsec:derivs-details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%