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]\cite{leal-u235-1999} and Fr\"{o}hner and Bouland [FF01]\cite{frohner_bouland_2001}.

Any number of additional possibilities exist for approximating the contribution of the
external resonances to the tail of the R-matrix. A logarithmic parameterization of the R-function is implemented in SAMMY; note that this is properly denoted as a function rather than a matrix, because it is diagonal with respect to the channels. The form used in the code is

\begin{equation}
    \begin{aligned}
        R_{c}^{e x t}(E)=\left.\bar{R}\right|_{\text {con, },} &+\bar{R}_{\text {lin, },} E+\bar{R}_{q, c} E^{2}-s_{\text {lin, },}\left(E_{c}^{u p}-E_{c}^{\text {down }}\right) \\
        &-\left(s_{\text {con, }, c}+s_{\text {lin, },} E\right) \ln \left[\frac{E_{c}^{u p}-E}{E-E_{c}^{\text {down }}}\right] .
    \end{aligned}
\end{equation}

Any or all of the seven free parameters may be varied during a SAMMY analysis (see Table VI B.2(\ref{table:format-par}), card set 3, and card set 3a). Note that $R_{c}^{ext}$ is strictly real in this parameterization.

The {\em u}-parameters (i.e., the parameters on which Bayes' equations will operate, as described in Section IV.C\ref{sec:construct-par-set}) associated with the external R-function are given by

\begin{equation}
    \begin{aligned}
        &u\left(E_{c}^{\text {down }}\right)=E_{c}^{\text {down }} \quad u\left(E_{c}^{u p}\right)=E_{c}^{u p}\\
        &u\left(\bar{R}_{\text {con }, c}\right)=\bar{R}_{\text {con }, c} \quad u\left(\bar{R}_{l i n, c}\right)=\bar{R}_{l i n, c} u\left(\bar{R}_{q, c}\right)=\bar{R}_{q, c}\\
        &u\left(s_{c o n, c}\right)=\sqrt{s_{c o n, c}} \quad u\left(s_{l i n, c}\right)=s_{l i n, c}
    \end{aligned}
\end{equation}

Of the current ENDF formats [ENDF-102]\cite{endf8}, only new LRF = 7 format permits this type of parameterization of the R-function. The more commonly used LRF = 3 format (the so-called Reich-Moore format) allows only the dummy-resonance option.

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

While SAMMY does not yet have the ability to calculate the full (unapproximated) R-matrix
of Lane and Thomas [AL58]\cite{lane_thomas_1958}, it is possible to use the Reich-Moore approximation in such a way that it mimics the full R-matrix with a high degree of accuracy. This is necessary, for example, in cases where there are interference effects between the (incident) neutron channel and a gamma channel - that is, for some low-mass nuclides.

The Reich-Moore approximation involves an aggregate treatment (``eliminated channels'') for the gamma widths (capture widths). Therefore, to approximate the full R-matrix, one sets the Reich-Moore gamma width to a very small number and uses an exit channel to define the actual gamma channel:

\begin{enumerate}
    \item Set the SAMMY gamma-channel widths to a very small number, perhaps 0.001.
    \item Define an exit channel to be the actual capture channel and assign appropriate values for the widths. Quantum numbers for this channel will be the same as those for fission channels (in particular, set LPENT = 0).
    \item When calculating capture cross sections, set the IFEXCL flag to 1 for all other (non-gamma) exit channels. (See ``Specifying individual reaction types'' above and card set 10.1 or 10.2 of Table VIA.1(\ref{table:format-inp}) for details.) When calculating other reaction cross sections, set the IFEXCL flag to 0 for the reaction channels of interest, to 1 for the capture channels, and to 1 for any other reaction channels to be excluded.
\end{enumerate}

When utilizing this option, SAMMY users should take care that results are not unduly influenced by the approximation in step 1 above. To test this, make radical changes in the value used for the gamma widths (e.g., set the value to 100.0 or 10$^{-6}$) and recalculate the cross section. Note that it is not possible to set these values to zero; doing so results in numerical overflow problems (because computers do not know how to calculate zero divided by zero).

% >>> NOTE: We should update this: maybe latest R-matrix code-comparison thru IAEA?
Comparisons between cross sections calculated by SAMMY and those generated by the R-matrix code EDA [GH75]\cite{hale-eda-comparison} using the same R-matrix parameters have shown agreement to $\approx 5$ significant digits [INDC03]\cite{indc-2004}. Some of the runs for those comparisons are now assembled into SAMMY test case tr125.

Test case tr110 shows an artificial but extreme example of a situation in which use of the Reich-Moore approximation gives very different results from those obtained via the full R-matrix. For this example, there are two resonances with parameter values as shown in Table \ref{table:pseudo-full-r-matrix}; plots of the curves calculated with those parameters are shown in Figure \ref{fig:pseudo-full-r-matrix}. As evident from the figure, the Reich-Moore curve lies between the two extreme R-matrix curves which show constructive (dashed curve) and destructive (dot-dash curve) interference.

\begin{table}[hbt!]
\centering
\caption{Parameter values used to illustrate Reich-Moore vs. full R-matrix calculations}
\label{table:pseudo-full-r-matrix}
    \begin{tabular}{p{50mm}|p{5mm}|p{20mm}|p{20mm}|p{20mm}|p{18mm}}
        \hline\hline
                                  & $\lambda$ & Energy (MeV) & $\overline{\Gamma}_{\lambda\gamma}$(eV) & $\Gamma_{\lambda n}$(eV) & Sign $\times\Gamma_{\lambda\gamma}$(eV)$^a$ \\ \hline\hline
        Reich-Moore               &  1  &  1.0  & 1.0       & 10000.0 &      \\ 
                                  &  2  &  1.1  & 1.1       & 11000.0 &      \\ \hline
        Pseudo-full R-matrix \# 1 &  1  &  1.0  & 10$^{-8}$ & 10000.0 & 1.0  \\ 
                                  &  2  &  1.1  & 10$^{-8}$ & 11000.0 & 1.1  \\ \hline
        Pseudo-full R-matrix \# 2 &  1  &  1.0  & 10$^{-8}$ & 10000.0 & 1.0  \\ 
                                  &  2  &  1.1  & 10$^{-8}$ & 11000.0 & -1.1 \\ \hline\hline
    \end{tabular}
    \footnotesize{$^a$Remember that the value given in the SAMMY PARameter file is not the partial width $\Gamma$ (which is always a positive number); rather, it is the sign of the reduced-width amplitude $\gamma$ multiplied by the partial width $\Gamma$. Hence, the negative sign in the final entry of this table is actually associated with the reduced-width amplitude for the capture channel. See Section II.B.1 \ref{subsec:r-matrix-RM} for further discussion.}
\end{table}

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/pseudo-full-r-matrix.pdf}
    \caption{Reich-Moore approximation vs. full R-matrix for artificial example of test case tr110.}
    \label{fig:pseudo-full-r-matrix}
\end{figure}

\noindent
\textbf{Different treatments for different capture channels}

Occasionally it may be convenient to treat certain gamma widths individually while treating most gamma widths in aggregate fashion. This can be accomplished by defining ``particle'' channels for the individual widths (as described above) and using the Reich-Moore capture channel (eliminated width) for the aggregate width.

To calculate the capture cross section in this situation, it is not sufficient to specify the data type as ``\texttt{CAPTURE}'', because that would give only the contribution from the aggregate width. To obtain the contribution from the individual widths, specify the data type as ``\texttt{REACTION}'' or (preferably) as ``\texttt{FINAL state pairs =}'' followed by the exact names specified for the gamma-channel particle-pairs. (See card sets 4 and 8 of Table VI A.1\ref{table:format-inp} and Section II.B.1.c \ref{subsec:r-matrix-RM} ``Specifying individual reaction types'' for details.)

To calculate the complete capture cross section, use ``\texttt{FINAL state pairs =}'' for the data type and add the command line

\texttt{ADD ELIMINATED CAPTUre channel to final state}

\noindent
This will cause SAMMY to add the contributions from the individual capture channels plus the
contribution from the aggregate channels.

The formula used to calculate the capture cross section is similar to Eq. (II B1 a.6)\ref{eq:sig-cap-in-terms-of-X}, with only the non-capture exit channels included in the summation over $c'$:

\begin{equation}
    \sigma_{\text {capture }}(E)=\frac{4 \pi}{k_{\alpha}^{2}} \sum_{J} g_{J \alpha} \sum_{c}\left[X_{c c}^{i}-\sum_{\substack{c^{\prime}=\text { non-capture } \\ \text { exit channels }}}\left\{X_{c c^{\prime}}^{i}+X_{c c^{\prime}}^{r}{ }^{2}\right\}\right].
\end{equation}

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

In addition to the preferred Reich-Moore formalism, SAMMY also offers the option to calculate cross sections using either the multilevel Breit-Wigner (MLBW) or the single-level Breit-Wigner (SLBW) [GB36]\cite{breit1936capture} approximation. These approximations have the advantage that the calculation occurs more rapidly because fewer computations are required; however, they also have the disadvantage that unphysical cross sections may be generated. Use of these options is discouraged for new analyses; the options are included within SAMMY for the sake of completeness, to permit use of SAMMY with most ENDF resonance parameter information, and to facilitate comparisons with older codes such as SIOB [GD78]\cite{desaussure-1978}.

% >>> removing mention of specific section
Formulae for MLBW and SLBW cross sections which are presented in this section are identical to those used in ENDF files [ENDF-102]\cite{endf8}, although they are not necessarily programmed in this fashion. Formulae for derivatives are given in Section II.D.2 \ref{subsec:derivs-BW}.

\textbf{The reader should be aware} that the ENDF version of MLBW (and hence, SAMMY's version of MLBW) does not correspond to the usual definition of multilevel Breit Wigner. Instead, only the elastic cross section is calculated with the multilevel formula; other partial cross sections for the MLBW format \textbf{are actually single level}.

A note regarding broadening: Historically, the Breit-Wigner formulations had the great advantage that the cross sections could be Doppler broadened analytically, using the high-energy approximation to the free-gas model of Doppler broadening (Section III.B.3)\ref{sec:doppler-broadening}. Results were written in terms of $\chi$ and $\psi$ functions, and computation was relatively rapid. However, with the advent of modern computers, more accurate cross sections and more accurate Doppler broadening computations can be accomplished rapidly, without resorting to these rather crude approximations. In SAMMY, Doppler and resolution broadening are accomplished numerically, in the same manner for MLBW and SLBW cross sections as for Reich-Moore cross sections, as described in Section III\ref{ch:exp-conditions} of this manual.

\newpage
\noindent
\textbf{Single and multilevel Breit-Wigner cross sections}

The MLBW elastic (scattering) cross section may be written in the form

\begin{equation}\label{eq:mlbw-elastic}
    \begin{aligned}
        \sigma^{\text {elastic }}=& \frac{\pi}{k^{2}} \sum_{J} g_{J} \sum_{c}\left\{(1-\cos 2 \varphi)\left(2-\sum_{\lambda} \Gamma_{\lambda c} \Gamma_{\lambda} / d_{\lambda}\right)\right.\\
        &+2 \sin 2 \varphi \sum_{\lambda} \Gamma_{\lambda c}\left(E-E_{\lambda}\right) / d_{\lambda} \\
        &\left.+\left(\sum_{\lambda} \Gamma_{\lambda c}\left(E-E_{\lambda}\right) / d_{\lambda}\right)^{2}+\left(\sum_{\lambda} \Gamma_{\lambda c} \Gamma_{\lambda} / 2 d_{\lambda}\right)^{2}\right\},
    \end{aligned}
\end{equation}

\noindent
in which the summation over $c$ includes only incident (i.e., neutron) channels. For SLBW, the level-level interference terms in this equation are dropped; that is, the summations over $\lambda$ in the last line are outside, rather than inside, the parentheses. The total width $\Gamma_\lambda$ in Eq. (II B3 a.1)\ref{eq:mlbw-elastic} is given by

\begin{equation}
    \Gamma_\lambda = \sum_c\Gamma_{\lambda c} + \overline{\Gamma}_{\lambda\gamma},
\end{equation}


\noindent 
% >>> adding a subscript gamma to lower-case gamma in text below
in which the sum over $c$ includes all particle channels (i.e., over all channels except the eliminated capture channel). Partial widths $\Gamma_{\lambda c}$ and $\overline{\Gamma}_{\lambda\gamma}$ are related to amplitudes $\gamma_{\lambda c}$ and $\overline{\gamma}_{\lambda\gamma}$, as in the Reich- Moore approximation, by

\begin{equation}
    \begin{aligned}
        \Gamma_{\lambda c}^{\text {neutron }} &=2 \gamma_{\lambda c}^{2} P_{c} \\
        \Gamma_{\lambda c}^{f i s s i o n} &=2 \gamma_{\lambda c}^{2} \\
        \text { and } \qquad\; \bar{\Gamma}_{\lambda \gamma} &=2 \bar{\gamma}_{\lambda \gamma}^{2} .
    \end{aligned}
\end{equation}

\noindent
(Note that we have again adopted the convention that the gamma channel be denoted by a bar over the symbol, even though it is not really treated differently from particle channels in the Breit Wigner approximations.) The denominator $d_\lambda$ in Eq. (II B3 a.1)\ref{eq:mlbw-elastic} represents

\begin{equation}\label{eq:d-lambda}
    d_\lambda = (E-E_\lambda)^2 + (\Gamma_\lambda/2)^2.
\end{equation}

For both MLBW and SLBW, the fission cross section is given by

\begin{equation}\label{eq:mlbw-fission}
    \sigma^{fission} = \frac{\pi}{k^2}\sum_Jg_J\sum_c\sum_{c'}\sum_\lambda \frac{\Gamma_{\lambda_c}\Gamma_{\lambda c'}}{d_\lambda},
\end{equation}

\noindent
% >>> removed reference to section for indiv. reax types
in which the sum over $c$ includes only incident (neutron) channels, $d_\lambda$ is again given by Eq. (II B3 a.4)\ref{eq:d-lambda}, and the sum over $c'$ includes all exit channels. Caution: In principle, Eq. (II B3 a.5)\ref{eq:mlbw-fission} could be used to describe any reaction channel, where term ``reaction'' encompasses any non-elastic, non-capture channel. However, the only reaction channel permitted in ENDF is fission; for SLBW only one fission channel is permitted, and for MLBW two fission channels may be used. In addition, ENDF allows only one neutron channel (i.e., only one entrance channel). Because SAMMY's Breit-Wigner options were created solely for use with ENDF evaluations (for comparison purposes), similar restrictions apply to the use of the Breit-Wigner approximations in SAMMY. (For the more general case involving other reactions such as inelastic, (n,p), (n,$\alpha$), or fission with more than two channels, use the Reich-Moore approximation as discussed in Section \ref{subsec:r-matrix-RM} ``Specifying individual reaction types'')

The Breit-Wigner form for the capture cross section is

\begin{equation}
    \sigma^{\text {capture }}=\frac{\pi}{k^{2}} \sum_{J} g_{J} \sum_{c} \sum_{\lambda} \frac{\Gamma_{\lambda c} \bar{\Gamma}_{\lambda \gamma}}{d_{\lambda}}
\end{equation}

\noindent
where, again, the sum over $c$ includes only incident (neutron) channels. Total and absorption cross sections are given by the appropriate sums of the other three cross sections,

\begin{equation}
    \sigma^{total} = \sigma^{elastic} + \sigma^{fission} + \sigma^{capture}
\end{equation}

\noindent
and

\begin{equation}
    \sigma^{absorption} = \sigma^{fission} + \sigma^{capture}.
\end{equation}

As noted in Section IV.C\ref{sec:construct-par-set}, it is the $u$-parameters on which Bayes' equations operate. The uparameters associated with the MLBW and SLBW resonances are defined similarly to those for Reich-Moore resonances:

\begin{equation}
    u(E_\lambda) = \pm \sqrt{|E_\lambda|},
\end{equation}

\noindent
where the negative sign is chosen if $E_\lambda <0$,

\begin{equation}
    u(\Gamma_{\lambda c}) = \gamma_{\lambda c}
\end{equation}

\noindent
and

\begin{equation}
    u(\overline{\Gamma}_{\lambda\gamma}) = \overline{\gamma}_{\lambda\gamma} \;\;.
\end{equation}

\noindent
(The reduced-width amplitudes and $\gamma_{\lambda c}$ and $\overline{\gamma}_{\lambda\gamma}$ may be either positive or negative. However, the sign is irrelevant in the Breit-Wigner equations, for which the reduced-width amplitudes enter only as squared quantities.)

The matching radius $a_c$ may also be varied (i.e., treated as a $u$-parameter) with the Breit-Wigner approximations.

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

An externally generated direct capture component may be added to the appropriate cross
section types (capture, absorption, and total) by including the phrase

\texttt{ADD DIRECT CAPTURE Component to capture, total, and absorption cross section}

\noindent
in the alphanumeric command section of the INPut file. When this command is present, the direct capture component for at least one of the nuclides is provided as a numerical function of energy, in a separate file (the ``DRC file''). SAMMY will linearly interpolate as needed between the energy points given.

The format of the DRC file is as follows:
\begin{enumerate}
    \item \textbf{First line:} key word ``NUClide Number'', followed by an equal sign ``='', followed by the nuclide number as specified in the PARameter file.
    \item \textbf{Second line:} energy (eV), value of direct capture component (barn), in 2F20 format.
    \item \textbf{Third line:} repeat second line as many times as needed.
    \item \textbf{Last line:} blank.
\end{enumerate}

These lines may be repeated for each nuclide as needed. Not all nuclides need to be included, but those which are included should be given in the same order as in the PARameter file. (For example, give the direct capture component for nuclides number 2, 4, and 7, rather than 4, 7, and 2.)

The actual value of the direct capture component added to the capture (and total and absorption) cross section for any given nuclide is the product of the value determined from the DRC file and a constant (energy-independent) coefficient whose value is specified as miscellaneous parameter DRCAP. See Table VI B.2 \ref{table:format-par} for details.

Test case tr076 contains examples.

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

\textbf{Note: Subject to change with inclusion of AMPX R-matrix engine!!!} even though the first iteration should provide the same results.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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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