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\chapter{Scattering Theory}\label{ch:scattering_theory}
Details of scattering theory have been well understood since the middle of the previous century, when they were summarized in a review article by Lane and Thomas [AL58]\cite{lane_thomas_1958}. A wealth of additional reference material is available to the student of scattering theory; only a few are listed here. The text by Foderaro [AF71]\cite{foderaro_1971} provides a more elementary introduction to the subject. One publication by Fr\"{o}hner [FF80]\cite{frohner_1978} is based on lectures presented at the International Centre for Theoretical Physics (ICTP) Winter Courses on Nuclear Physics and Reactors, 1978; this is a comprehensive and useful guide to applied neutron resonance theory. It includes a variety of topics, including preparation of data, various approximations to scattering theory, Doppler broadening, experimental complications, data-fitting procedures, and statistical tests. Another Fr\"{o}hner paper [FF00]\cite{jeff18_frohner} is somewhat more theoretical, and covers many aspects of data fitting in the resonance region.
The particular aspect of scattering theory with which we are concerned is the R-matrix formalism. A summary of the underlying principles is given here.
R-matrix theory is a mathematically rigorous phenomenological description of what is actually seen in an experiment (i.e., the measured cross section). The theory is not a model of neutron-nucleus interaction, in the sense that it makes no assumptions about the underlying physics of the interaction. Instead it parameterizes the measurement in terms of quantities such as the interaction radii and boundary conditions, resonance energies and widths, and quantum numbers; values for these parameters may be determined by fitting theoretical calculations to observed data. The theory is mathematically correct, in that it is analytic, unitary, and rigorous; nevertheless, in practical applications, the theory is always approximated in some fashion.
R-matrix theory is based on the following assumptions\footnote{In practical applications two of these four assumptions may be violated in one degree or another: (1) The theory may be used for relativistic neutron energies, and corrected for relativistic effects; nevertheless, non-relativistic quantum mechanics is assumed. (2) A fission experiment with more than two final products is treated as a two-step process. That is, the immediate result of the neutron-nuclide interaction is assumed to be limited to two final products, at least one of which decays prior to detection.}:
(1) the applicability of non-relativistic quantum mechanics;
(2) the absence or unimportance of all processes in which more than two product nuclei are formed;
(3) the absence or unimportance of all processes of creation or destruction; and
(4) the existence of a finite radial separation beyond which no nuclear interactions occur, although Coulomb interactions are given special treatment.
R-matrix theory is expressed in terms of channels, where a channel is defined as a pair of (incoming or outgoing) particles, plus specific information relevant to the interaction between the two particles. A schematic depicting entrance and exit channels is shown in Fig. \ref{scattering-theory_rmatrix_channel_diagram}. Note that entrance channels can also occur as exit channels, but some exit channels (e.g., fission channels) do not occur as entrance channels. Two interacting particles are shown in the portion of the figure that is labeled ``Interior Region''; here the particles are separated by less than the interaction radius $a$.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{figures/rmatrix_inc_channel.pdf} \\[\smallskipamount]
\includegraphics[width=0.4\textwidth]{figures/rmatrix_interior_region.pdf}\hfill
\includegraphics[width=0.4\textwidth]{figures/rmatrix_exit_channel.pdf}
\caption{Schematic of entrance and exit channels as used in scattering theory. For the interior region (with separation distance $r < a$), no assumptions are made about the nature of the interaction. In the figure, $m$, $i$, and $z$ refer to the mass, spin, and charge of the incident particle while $M$, $I$ and $Z$ refer to the target particle. Orbital angular momentum is denoted by $l$ and velocity by $v$. Primes are used for post-collision quantities.}
\label{scattering-theory_rmatrix_channel_diagram}
\end{figure}
In Section \ref{sec:equations-for-scattering-theory}, general equations of scattering theory are presented and their derivations discussed. The fundamental R-matrix equations are presented. Section II.A.1\ref{subsec:r-matrix-a-matrix} gives a detailed derivation of the equations for a simple case. Section II.A.2\ref{subsec:derivation-of-scat-theory} shows the relationship between the R-matrix and the A-matrix, which is another common representation of scattering theory.
The approximations to R-matrix theory available in the SAMMY code are detailed in Section II.B\ref{sec:versions-of-r-matrix}. The recommended choice for most applications is the Reich-Moore approximation, described in Section II.B.1\ref{subsec:r-matrix-RM}. For some applications, the Reich-Moore approximation is inadequate; for those cases, a method for using SAMMY's Reich-Moore approximation to mimic the full (exact) R-matrix is presented Section II.B.2\ref{subsec:r-matrix-full}. Two historically useful but now obsolete approximations are single-level and multilevel Breit Wigner (SLBW and MLBW), discussed in Section II.B.3\ref{subsec:r-matrix-BW}. Provisions for including non-compound (direct) effects are discussed in Section II.B.4\ref{subsec:direct-capture}.
In Section II.C\ref{sec:details-and-conventions}, details are given for the SAMMY nomenclature and other conventions, for transformations to the center-of-momentum system, and for the calculation of penetrability, shift factors, and hard-sphere phase shifts in both Coulomb and non-Coulomb cases.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Equations For Scattering theory} \label{sec:equations-for-scattering-theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, equations for scattering theory are presented but not derived. Specifics for the R-matrix formulation of scattering theory are presented in Section II.A.1\ref{subsec:r-matrix-a-matrix}, which provides a discussion of an alternative formulation (the A-matrix). Readers interested in the derivation of the equations for scattering theory are referred to the Lane and Thomas article \cite{lane_thomas_1958} for a detailed derivation in the general case, or to Section II.A.2\ref{subsec:derivation-of-scat-theory} of this document for a simplified version.
In scattering theory, a channel may be defined by $c = (\alpha, l, s, J)$, where the following definitions apply:
\begin{itemize}
\item $\alpha$ represents the two particles making up the channel; $\alpha$ includes mass ($m$ and $M$), charge ($z$ and $Z$), spin ($i$ and $I$ ) with associated parities, and all other quantum numbers for each of the two particles, plus the Q-value (equivalent to the negative of the threshold energy in the
center of momentum system).
\item $l$ is the orbital angular momentum of the pair, and the associated parity is given by $(-1)^l$.
\item $s$ represents the channel spin (including the associated parity); that is, $s$ is the quantized
vector sum of the spins of the two particles of the pair: $\vec{s} = \vec{i} + \vec{I}$
\item $J$ is the total angular momentum (and associated parity); that is, $J$ is the quantized vector sum
of $l$ and $s$: $\vec{J} = \vec{l} + \vec{s}$.
\end{itemize}
\noindent
Only $J$ and its associated parity $\pi$ are conserved for any given interaction. The other quantum numbers may differ from channel to channel, as long as the sum rules for spin and parity are obeyed. Within this document and within the SAMMY code, the set of all channels with the same $J$ and $\pi$ are called a ``spin group.''
In all formulae given below, spin quantum numbers (e.g., $J$ ) are implicitly assumed to include the associated parity. Quantized vector sum rules are implicitly assumed to be obeyed. Readers unfamiliar with these sum rules are referred to Section II.C.1.a\ref{} for a mini-tutorial on the subject.
Let the angle-integrated cross sections from entrance channel $c$ to exit channel $c'$ with total angular momentum $J$ be represented by $\sigma_{cc'}$. This cross section is given in terms of the scattering matrix $U_{cc'}$ as
\begin{equation}\label{eq:sigma-ccprime}
\sigma_{cc'} = \frac{\pi}{k_\alpha^2}g_{J\alpha}\left|e^{2iw_c}\delta_{cc'}-U_{cc'}\right|^2\delta_{JJ'} \:,
\end{equation}
\noindent
where $k_\alpha$ is the wave number (and $K_\alpha = \hbar k_\alpha =$ center-of-mass momentum) associated with incident particle pair $\alpha$, $g_{J\alpha}$ is the spin statistical factor, and $w_c$ is the Coulomb phase-shift difference. Note that $w_c$ is zero for non-Coulomb channels. (Details for the charged-particle case are presented in Section II.C.4.\ref{subsec:charged-particle-conventions}) The spin statistical factor $g_{J\alpha}$ is given by
\begin{equation}\label{eq:spin-stat-factor}
g_{J\alpha} = \frac{2J+1}{(2i+1)(2I+1)} \:,
\end{equation}
\noindent
and center-of-mass momentum $K_\alpha$ by
\begin{equation}\label{eq:center-of-mass-momentum}
K_\alpha^2 = (\hbar k_\alpha)^2 = \frac{2mM^2}{(m+M)^2}E \:.
\end{equation}
\noindent
Here $E$ is the \textbf{laboratory} kinetic energy of the incident (moving) particle. A derivation of this value for $K_\alpha$ is given in Section II.C.2\ref{subsec:kinematic-conventions}.
The scattering matrix $U$ can be written in terms of matrix $W$ as
\begin{equation}\label{eq:scat-matrix}
U_{cc'} = \Omega_cW_{cc'}\Omega_{c'} \:,
\end{equation}
\noindent
where $\Omega$ is given by
\begin{equation}\label{eq:omega}
\Omega_c = e^{i(w_c-\phi_c)} \:.
\end{equation}
\noindent
Here again, $w_c$ is zero for non-Coulomb channels, and the potential scattering phase shifts for non-Coulomb interactions $\phi_c$ are defined in many references (e.g., \cite{lane_thomas_1958}) and shown in Table \ref{tab:penetrabilities}. The matrix $W$ in Eq. \ref{eq:scat-matrix} is related to the R-matrix (in matrix notation with indices suppressed) via
\begin{equation}\label{eq:W-matrix}
W = P^{1/2}\left(I-RL\right)^{-1}\left(I-RL^*\right)P^{-1/2} \:.
\end{equation}
\noindent
The quantity $I$ in this equation represents the identity matrix, and superscript $*$ indicates a complex conjugate. The form of the R-matrix is given in Section II.A.1\ref{subsec:r-matrix-a-matrix} in general Section II.B\ref{sec:versions-of-r-matrix} for the versions used in SAMMY. The quantity $L$ in Eq. \ref{eq:W-matrix} is given by
\begin{equation}\label{eq:L-matrix}
L = (S-B) + iP \:,
\end{equation}
\noindent
with $P$ being the penetration factor (penetrability) $S$ the shift factor, and $B$ the arbitrary boundary constant at the channel radius $a_c$. $P$ and $S$ are functions of energy $E$, and also depend on the orbital angular momentum $l$ and the channel radius $a_c$. Formulae for $P$ and $S$ are found in many references (see, for example Eq. (2.9) in \cite{lynn_1968}).
For non-Coulomb interactions, the penetrability and shift factor have the form
\begin{equation}\label{eq:pen-shift-func-of-rho}
P\rightarrow P_l(\rho) \qquad \text{and} \qquad S \rightarrow S_l(\rho) \:,
\end{equation}
\noindent
where $\rho$ is related to the center-of-mass momentum which in turn is related to the laboratory energy of the incident particle $(E)$. For arbitrary channel $c$ with a particle pair $\alpha$, orbital angular momentum $l$, and channel radius $a_c$, $\rho$ has the form
\begin{equation}\label{eq:de-Broglie-radius}
\rho = k_\alpha a_c = \frac{1}{\hbar} \sqrt{\frac{2m_\alpha M_\alpha}{m_\alpha+M_\alpha} \frac{M}{m+M}} \sqrt{(E-\Xi_\alpha)}\: a_c \:,
\end{equation}
\noindent
as shown in Section II.C.2 \ref{subsec:kinematic-conventions}. Here $\Xi_\alpha$ is the energy threshold for the particle pair $\alpha$, $m_\alpha$ and $M_\alpha$ are the masses of the two particles of particle pair $\alpha$, and $m$ and $M$ are the masses of the incident particle and target nuclide, respectively.
Appropriate formulae\footnote{To avoid ambiguity, it should be stated that below the channel threshold, that is, for $(E-\Xi_\alpha)<0$, SAMMY uses the convention of Lane-Thomas, namely setting $P_c=0$ and $S_c=\text{Re}(L_c)=L_c$, instead of using an analytical continuation of the shift and penetrability function in the complex plane by computing the expressions in Table \ref{tab:penetrabilities} as a function of an imaginary $\rho$ for which $iP_c(\rho)$ becomes real-valued, but separate from $S_c(\rho)$.} for $P$, $S$, and $\phi$ in the non-Coulomb case are shown in Table II.A.1 \ref{tab:penetrabilities}. For two charged particles, formulae for the penetrabilities are given in Section II.C.4 \ref{subsec:charged-particle-conventions}. The energy dependence of fission and capture widths is negligible over the energy range of these calculations. Therefore, a penetrability of unity may be used.
% multiline cell: \begin{tabular}{@{}c@{}} line1 \\ line2 \end{tabular}
\begin{threeparttable}
\centering
\caption{Hard-sphere penetrability (penetration factor) $P$, level shift factor $S$, and potential-scattering phase shift $\phi$ for orbital angular momentum $l$, wave number $k$, and channel radius $a_c$, with $\rho=ka_c$.} \label{tab:penetrabilities}
\begin{tabular}{ l c c c } \hline\hline
$\mathbf{l}$ & $\mathbf{P_l}$ & $\mathbf{S_l}$ & $\mathbf{\phi_l}$ \cr
\hline\hline
0 & $\rho$ & 0 & $\rho$ \cr
1 & $\rho^3/(1+\rho^2)$ & $-1/(1+\rho^2)$ & $\rho-\tan^{-1}\rho$ \cr
2 & $\rho^5/(9+3\rho^2+\rho^4)$ & $-(18+3\rho^2)/(9+3\rho^2+\rho^4)$ & $\rho-\tan^{-1}\left[3\rho/(3-\rho^2)\right]$ \cr
3 & \begin{tabular}{@{}c@{}}$\rho^7/$ \\ $(225+45\rho^2+6\rho^4+\rho^6)$\end{tabular} & \begin{tabular}{@{}c@{}} $-(675+90\rho^2+6\rho^4)/$ \\ $(225+45\rho^2+6\rho^4+\rho^6)$ \end{tabular} & $\rho-\tan^{-1}\left[\rho(15-\rho^2)/(15-6\rho^2)\right]$ \cr
\hline
$l$ & $\frac{\rho^2P_{l-1}}{(1-S_{l-1})^2+P_{l-1}^2}$ & $\frac{\rho^2(l-S_{l-1})}{(1-S_{l-1})^2+P_{l-1}^2}-l$ & $\phi_{l-1} - \tan^{-1}\left(P_{l-1}/(l-S_{l-1})\right)$\tnote{\textdagger} \cr
\hline\hline
\end{tabular}
\begin{tablenotes}
\item[\textdagger] \footnotesize{The iterative formula for $\phi_l$ could also be defined by $B_l = (B_{l-1}+X_l)/(1-B_{l-1}X_l)$ where $B_l=\tan(\rho-\phi_l)$ and $X_l=P_{l-1}/(l-S_{l-1})$}
\end{tablenotes}
\end{threeparttable}\\
Formulae for a particular cross section type can be derived by summing over the terms in Eq. (II A.1)\ref{eq:sigma-ccprime}. For the total cross section, the sum over all possible exit channels and all spin groups gives
\begin{equation}\label{eq:sigma-tot}
\begin{aligned}
\sigma^{total} & = \sum_{\substack{\text{incident} \\ \text{channels} \\ c }} \sum_{\substack{\text{all} \\ \text{channels} \\ c' }} \sum_J \frac{\pi}{k_\alpha^2}g_\alpha|\delta_{cc'}-U_{cc'}|^2 \\
& = \frac{\pi}{k_\alpha^2} \sum_{J}g_J \sum_{\substack{\text{incident} \\ \text{channels} \\ c }} \sum_{\substack{\text{all} \\ \text{channels} \\ c' }} \left( \delta_{cc'}-U_{cc'}\delta_{cc'}-U^*_{cc'}\delta_{cc'}+\left|U_{cc'}\right|^2 \right) \\
& = \frac{2\pi}{k_\alpha^2} \sum_J g_J \sum_{\substack{\text{incident} \\ \text{channels} \\ c }} \left( 1-\text{Re}\left(U_{cc}\right) \right).
\end{aligned}
\end{equation}
\noindent
For non-charged incident particles, the elastic (or scattering) cross section is given by
\begin{equation}\label{eq:sigma-scat}
\sigma_{\alpha\alpha} = \frac{\pi}{k_\alpha^2} \sum_J g_J \sum_{\substack{ c=\text{incident} \\ \text{channel} }} \left( 1-2\text{Re}\left(U_{cc}\right) + \sum_{\substack{c'=\text{incident} \\ \text{channel} }} \left|U_{cc'}\right|^2 \right).
\end{equation}
\noindent
Similarly, the cross section for any non-elastic reaction can be written
\begin{equation}\label{eq:sigma-reaction}
\sigma_\alpha^{reaction} = \frac{\pi}{k_\alpha^2} \sum_J g_J \sum_{\substack{ c=\text{incident} \\ \text{channel} }} \sum_{\substack{ c'=\text{reaction} \\ \text{channel} }} \left| U_{cc'} \right|^2.
\end{equation}
\noindent
In particular, the capture cross section could be written as the difference between the total and all
other cross sections,
\begin{equation}\label{eq:sigma-capture}
\sigma_\alpha^{reaction} = \frac{\pi}{k_\alpha^2} \sum_J g_J \sum_{\substack{ c=\text{incident} \\ \text{channel} }} \left( 1 - \sum_{\substack{ c'=\text{all channels} \\ \text{except capture} }} \left| U_{cc'} \right|^2 \right).
\end{equation}
\noindent
(This form will be used later, in Section II.B.1.a, when the capture channels are treated in an
approximate fashion.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{R-Matrix and A-Matrix Equations}\label{subsec:r-matrix-a-matrix}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The R-matrix was introduced in Eq. \ref{eq:W-matrix} as
\begin{equation} % don't ref, give \ref{eq:W-matrix}
W = P^{1/2}\left(I-RL\right)^{-1}\left(I-RL^*\right)P^{-1/2} \:,
\end{equation}
\noindent
but the formula for the R-matrix was not given there. If $\lambda$ represents a particular resonance (or energy level), then the general form for the R-matrix is
\begin{equation}\label{eq:r-matrix}
R_{cc'} = \sum_\lambda \frac{ \gamma_{\lambda c}\gamma_{\lambda c'} }{ E_\lambda - E } \delta_{J,J'} \:,
\end{equation}
\noindent
where $E_\lambda$ represents the energy of the resonance, and the reduced width amplitude $\gamma$ is related to the partial width $\Gamma$ by
\begin{equation}\label{eq:reduced-widths}
\Gamma_{\lambda c} = 2 \bf{P_c}\gamma_{\lambda c}^2.
\end{equation}
\noindent
Note that in Eq. \ref{eq:r-matrix} that energies and widths are given in laboratory frame of reference, while the derivation in \S \ref{subsec:derivation-of-scat-theory} is in center-of-mass (please see Eq. \ref{eq:lab-to-com-params} for the relationship of laboratory and center-of-mass parameters). The sum in Eq. (II A1.2)\ref{eq:r-matrix} contains an infinite number of levels. All channels, including the ``gamma channel'' for which one of the particles is a photon, are represented by the channel indices.
The R-matrix is not the only possibility for parameterization of the scattering matrix. In the R-matrix formulation, equations are expressed in terms of channel-channel interactions. It is also possible to formulate scattering theory in terms of level-level interactions; this formulation uses what is called the A-matrix, which is defined as
\begin{equation}\label{eq:a-matrix}
A_{\mu\lambda}^{-1} = \left( E_\lambda - E \right)\delta_{\mu\lambda} - \sum_c \gamma_{\mu c}L_c\gamma_{\lambda c} .
\end{equation}
\noindent
To see the relationship of the A-matrix to the R-matrix, we begin by multiplying both
sides of Eq. (II A1.4)\ref{eq:a-matrix} by $A$ and summing over $\lambda$:
\begin{equation}
\begin{aligned}
\sum_\lambda A_{\mu\lambda}^{-1}A_{\lambda\nu} & = \sum_\lambda \left(E_\lambda-E\right)\delta_{\mu\lambda}A_{\lambda\nu} - \sum_c \gamma_{\mu c} L_c \gamma_{\lambda c} A_{\lambda \nu}, \\
& \text{or} \\
\delta_{\mu\nu} & = \left(E_\mu-E\right)A_{\mu\nu} - \sum_c \gamma_{\mu c}L_c \sum_\lambda \gamma_{\lambda c}A_{\lambda\nu} .
\end{aligned}
\end{equation}
\noindent
Dividing by $(E_\mu - E)$, multiplying on the left by $\gamma_{\mu c'}$ and on the right by $\gamma_{\nu c''}$, and summing over $\mu$ puts this equation into the form
\begin{equation}\label{a-matrix-transform-1}
\begin{aligned}
\sum_\mu \gamma_{\mu c'}\left(E_{\mu}-E\right)^{-1}\delta_{\mu\nu}\gamma_{\nu c''} = & \sum_\mu \gamma_{\mu c'}\left(E_\mu-E\right)^{-1}\left(E_\mu-E\right)A_{\mu\nu}\gamma_{\nu c''} \\
& - \sum_\mu \gamma_{\mu c'} \left(E_{\mu}-E\right)^{-1} \sum_c \gamma_{\mu c} L_c \sum_\lambda \gamma_{\lambda c} A_{\lambda \nu}\gamma_{\nu c''} ,
\end{aligned}
\end{equation}
\noindent
which can be reduced to
\begin{equation}\label{a-matrix-transform-2}
\begin{aligned}
\gamma_{\nu c'}\left(E_{\nu}-E\right)^{-1}\gamma_{\nu c''} = & \sum_{\mu} \gamma_{\mu c'} A_{\mu\nu}\gamma_{\nu c''} \\
& - \sum_c \left[ \sum_\mu \gamma_{\mu c'}\left(E_{\mu}-E\right)^{-1}\gamma_{\mu c} \right] L_c \sum_\lambda \gamma_{\lambda c} A_{\lambda\nu} \gamma_{\nu c''}.
\end{aligned}
\end{equation}
\noindent
Summing over $\nu$ puts this into the form
\begin{equation}
\begin{aligned}
\left[ \sum_\nu \gamma_{\nu c'}\left(E_{\nu}-E\right)^{-1}\gamma_{\nu c''} \right] = & \sum_{\mu\nu} \gamma_{\mu c'} A_{\mu\nu}\gamma_{\nu c''} \\
& - \sum_c \left[ \sum_\mu \gamma_{\mu c'}\left(E_{\mu}-E\right)^{-1}\gamma_{\mu c} \right] L_c \sum_{\lambda\nu} \gamma_{\lambda c} A_{\lambda\nu} \gamma_{\nu c''},
\end{aligned}
\end{equation}
\noindent
in which we can replace the quantities in square brackets by the R-matrix, giving
\begin{equation}
\begin{aligned}
R_{c'c''} = & \sum_{\mu\nu} \gamma_{\mu c'} A_{\mu\nu}\gamma_{\nu c''} - \sum_c R_{c'c} L_c \sum_{\lambda\nu} \gamma_{\lambda c} A_{\lambda\nu} \gamma_{\nu c''}, \\
= & \sum_c \left[ \delta_{c'c} - R_{c'c}L_c \right] \sum_{\lambda\nu}\gamma_{\lambda c} A_{\lambda\nu} \gamma_{\nu c''}.
\end{aligned}
\end{equation}
\noindentSolving for the summation, this equation can be rewritten as
\begin{equation}\label{eq:r-matrix-to-a-matrix}
\left[(I-RL)^{-1}R\right]_{cc''} = \sum_{\lambda\nu} \gamma_{\lambda c} A_{\lambda\nu} \gamma_{\nu c''}.
\end{equation}
\noindent
To relate this to the scattering matrix, we note that Eq. (II A.6)\ref{eq:W-matrix} can be rewritten using Eq. (II A.7)\ref{eq:L-matrix} into the form
\begin{equation}\label{eq:w-matrix-transform}
\begin{aligned}
W & = P^{1/2}\left(I-RL\right)^{-1}\left(I-RL^*\right)P^{-1/2} \\
& = P^{1/2}\left(I-RL\right)^{-1}\left(I-RL+2iRP\right)P^{-1/2} \\
& = P^{1/2}\left[\left(I-RL\right)^{-1}\left(I-RL\right) + 2i\left(I-RL\right)^{-1}RP\right]P^{-1/2} \\
& = P^{1/2}P^{-1/2} + 2iP^{1/2}\left(I-RL\right)^{-1}RP P^{-1/2} \\
& = I + 2iP^{1/2}\left(I-RL\right)^{-1}RP^{1/2} .
\end{aligned}
\end{equation}
\noindent
Comparing Eq. (II A1.10)\ref{eq:r-matrix-to-a-matrix} to Eq. (II A1.11)\ref{eq:w-matrix-transform} gives, in matrix form,
\begin{equation}
W = I + 2iP^{1/2}\gamma A\gamma P^{1/2} .
\end{equation}
\noindent
These equations are exact; no approximations have been made.
One common approximation should be discussed here: the ``eliminated channel'' approximation, for which one particular type of channel is treated in aggregate and assumed to not interfere from level to level. This is most easily understood in the A-matrix definition, Eq. (II A1.4)\ref{eq:a-matrix}; assuming no level-level interference for the gamma channels (for example), this equation can be approximated as
\begin{equation}\label{eq:a-matrix-rm-approx}
A_{\mu\lambda}^{-1} \approx (E_\lambda-E)\delta_{\mu\lambda} - \left[\sum_{\substack{\gamma=\text{gamma} \\ \text{channels}}} \gamma_{\mu\gamma}L_{\gamma}\gamma_{\lambda\gamma}\right]\delta_{\mu\lambda} - \sum_{\substack{c=\text{particle} \\\text{channels}}} \gamma_{\mu c}L_c\gamma_{\lambda c} .
\end{equation}
\noindent
The quantity in square brackets corresponds to those channels for which the level-level
interference is to be neglected; that is, only the interactions within one level are important. For
gamma channels, $L=S+iP$ reduces to $L=i$, so Eq. (II A1.13)\ref{eq:a-matrix-rm-approx} becomes
\begin{equation}\label{eq:a-matrix-rm-reduced}
A_{\mu\lambda}^{-1} \approx \left(E_\lambda-E-i\overline{\Gamma}_{\lambda\gamma}/2\right)\delta_{\mu\lambda} - \sum_{\substack{c=\text{particle} \\ \text{channels}}} \gamma_{\mu c}L_c\gamma_{\lambda c} .
\end{equation}
\noindent
The bar over $\overline{\Gamma}_{\lambda\gamma}$ is used to indicate the special treatment for this channel.
In this form, our expression for $A$ is analogous to the exact expression in Eq. (II A1.4)\ref{eq:a-matrix} with two modifications: the additional imaginary term is added to the energy difference, and the sum over the channels includes only the ``particle channels'' (non-eliminated channels). It is therefore possible to immediately write the R-matrix formula for the eliminated-channel approximation as
\begin{equation}
R_{cc'} = \sum_{\lambda}\frac{\gamma_{\lambda c}\gamma_{\lambda c'}}{E_\lambda-E-i\overline{\Gamma}_{\lambda\gamma}/2} \delta_{JJ'} ,
\end{equation}
where the channel indices c and c ' refer only to particle channels, not to the gamma channels.
This formula for the R-matrix is the Reich-Moore approximation and is the form which is used in
the SAMMY code. See Section II.B.1 for more about this formulation of R-matrix theory.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Derivation of Scattering Theory Equations}\label{subsec:derivation-of-scat-theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Many authors have given derivations of the equations for the scattering matrix in terms of the R-matrix. Sources for the derivation shown here are unpublished lecture notes of Fr\"{o}hner [FF02]\cite{frohner_lecture_notes_02}, presented at the SAMMY workshop in Paris in 2002, and Foderaro [AF71]\cite{foderaro_1971}. This derivation is valid for only the simple case of spinless projectiles and target nuclei, assuming only elastic scattering and absorption. For the general case, the reader is referred to Lane and Thomas [AL58]\cite{lane_thomas_1958}.
\noindent
\textbf{Schr\"{o}dinger equation}
The Schr\"{o}dinger equation with a complex potential is
\begin{equation}
\left( \frac{-\hbar^2}{2\mu}\nabla^2 + V + iW \right)\psi = E^{\text{(CoM)}}\psi, \end{equation}
\noindent
in which one can consider that $V$ causes scattering and $W$ causes absorption and the reduced mass $\mu = \frac{mM}{m+M}$ ($m$ and $M$ defined as before). \textbf{Note that energies in \S \ref{subsec:derivation-of-scat-theory} are in the center-of-mass (CoM) frame.} The wave function
can be expanded in the usual fashion,
\begin{equation}
\psi\left(r,\text{cos}(\theta)\right) = \sum_{l=0}^\infty \frac{u_l(r)}{r} P_l(\text{cos}(\theta)) ,
\end{equation}
\noindent
for which the radial portion obeys the equation
\begin{equation}\label{eq:radial-wave-equation}
\frac{d^2u_l}{dr^2} + \left[ k^2-\frac{2\mu}{\hbar^2}(V+iW)-\frac{l(l+1)}{r^2} \right]u_l = 0,
\end{equation}
\noindent
subject to the conditions that $|\psi|^2$ is everywhere finite and that
\begin{equation}
u_l(r=0) = 0.
\end{equation}
In the external region, $r>a$, the nuclear forces are zero $(V=W=0)$, so the solution has
the form
\begin{equation}\label{eq:in-out-wave-funcs}
u_l(r) = I_l(r) - U_l O_l(r) .
\end{equation}
$I_l$ represents an incoming free wave, and $O_l$ represents an outgoing free wave. $U_l$ is the ``collision function'' or ``S function'' that describes the effects of the nuclear interaction, giving both the attenuation and the phase shift of the outgoing wave:
\begin{equation}
\begin{aligned}
|U_l|^2 = 1 & \text{ for } W = 0, \\
|U_l|^2 < 1 & \text{ for } W \neq 0.
\end{aligned}
\end{equation}
\noindent
Our goal is to determine an appropriate analytic form for $U_l$.
\noindent
\textbf{Orthogonal eigenvectors in interior region}
For the interior region $r<a$ , we define eigenfunctions $w_{\lambda l}(r)$ and eigenvalues $E_\lambda$,
\begin{equation}
E_\lambda^\text{(CoM)} = \frac{\hbar^2k_\lambda^2}{2\mu},
\end{equation}
\noindent
for the wave equation without absorption $(W = 0)$,
\begin{equation}\label{eq:wave-func-without-abs}
\frac{d^2w_{\lambda l}}{dr^2} + \left[k_\lambda^2-\frac{2\mu}{\hbar^2}V-\frac{l(l+1)}{r^2}\right]w_{\lambda l} = 0,
\end{equation}
\noindent
for which the boundary conditions are
\begin{equation}\label{eq:schrod-bound-conditions}
w_{\lambda l}(r=0) = 0 \;\;\;\;\; \text{and} \;\;\;\;\; \frac{a}{w_{\lambda l}(a)}\frac{dw_{\lambda l}}{dr}\Bigr|_{r=a} = B_l.
\end{equation}
\noindent
Note that $w_{\lambda l}(r)$ is real if the boundary parameter $B_l$ is chosen to be real. The eigenfunctions are orthogonal, since
\begin{equation}\label{eq:eigen-func-orthog} \begin{aligned}
\int_0^a\left( \frac{d^2w_{\lambda l}}{dr^2}w_{\mu l} - w_{\lambda l}\frac{d^2w_{\mu l}}{dr^2} \right) dr & = \int_0^a \frac{d}{dr} \left( \frac{dw_{\lambda l}}{dr}w_{\mu l} - w_{\lambda l}\frac{dw_{\mu l}}{dr} \right) dr \\
& = \left[\frac{dw_{\lambda l}}{dr}w_{\mu l} - w_{\lambda l}\frac{dw_{\mu l}}{dr}\right]_0^a \\
& = \frac{dw_{\lambda l}}{dr}\Bigr|_{r=a} w_{\mu l}(a) - w_{\lambda l}(a)\frac{dw_{\mu l}}{dr}\Bigr|_{r=a} - [0] \\
& = \frac{B_l}{a}\left[ w_{\lambda l}(a)w_{\mu l}(a)-w_{\lambda l}(a)w_{\mu l}(a) \right] = 0,
\end{aligned}
\end{equation}
\noindent
in which both equations of (II A2.9)\ref{eq:schrod-bound-conditions} have been invoked. The integral in Eq. (II A2.10)\ref{eq:eigen-func-orthog} can also be evaluated using Eq. (II A2.8)\ref{eq:wave-func-without-abs}, giving
\begin{equation}\label{eq:reform-orthog}
\begin{aligned}
\int_0^a\left( \frac{d^2w_{\lambda l}}{dr^2}w_{\mu l} - w_{\lambda l}\frac{d^2w_{\mu l}}{dr^2} \right) dr & = \int_0^a\left( \left[-k_\lambda^2-\frac{2\mu V}{\hbar^2}\right]w_{\lambda l}w_{\mu l} - w_{\lambda l}\left[-k_\lambda^2-\frac{2\mu V}{\hbar^2}\right]w_{\lambda l} \right)dr \\
& = \int_0^a\left( -k_\lambda^2w_{\lambda l}w_{\mu l} + k_\mu^2w_{\lambda l}w_{\mu l} \right)dr \\
& = -(k_\lambda^2-k_\mu^2)\int_0^a w_{\lambda l}w_{\mu l}dr .
\end{aligned}
\end{equation}
\noindent
Equating Eq. (II A2.10)\ref{eq:eigen-func-orthog} to Eq. (II A2.11)\ref{eq:reform-orthog} gives
\begin{equation}
(k_\lambda^2-k_\mu^2)\int_0^a w_{\lambda l}w_{\mu l}dr = 0.
\end{equation}
\noindent
For $\lambda \neq \mu$, assuming no degenerate states, it therefore follows that
\begin{equation}
\int_0^a w_{\lambda l}w_{\mu l}dr = 0 \;\;\;\;\; \text{if} \;\;\;\;\; \lambda\neq\mu.
\end{equation}
The orthogonality of the eigenvectors is therefore established. We assume that these wave
functions are normalized such that
\begin{equation}\label{eq:normalized-wave-func}
\int_0^a w_{\lambda l}w_{\mu l}dr = \delta_{\lambda\mu}.
\end{equation}
\noindent
\textbf{Matching at the surface}
The internal wave function for the true potential (including the imaginary part $iW$ ) can be
expanded in terms of the eigenfunctions as
\begin{equation}\label{eq:internal-wave-func}
u_l(r) = \sum_{\lambda} c_{\lambda l}w_{\lambda l}(r) \;\;\;\;\; \text{for} \;\; r\leq a,
\end{equation}
\noindent
with
\begin{equation}\label{eq:c-lambda-l}
c_{\lambda l} = \int_0^a u_l w_{\lambda l} dr.
\end{equation}
\noindent
This equation for $c_{\lambda l}$ is derived by multiplying Eq. (II A2.15)\ref{eq:internal-wave-func} by $u_{\lambda l}(r)$ , integrating, and applying Eq. (II A2.14)\ref{eq:normalized-wave-func}.
\noindent
Consider now the integral
\begin{equation}
\int_0^a\left( \frac{d^2u_{l}}{dr^2}w_{\lambda l} - u_{l}\frac{d^2w_{\lambda l}}{dr^2} \right) dr,
\end{equation}
\noindent
which can be expanded by use of Eqs. (II A2.3)\ref{eq:radial-wave-equation} and (II A2.8)\ref{eq:wave-func-without-abs} to give
\begin{equation}\label{eq:schroding-plugin}
\begin{aligned}
& \int_0^a \left( \frac{d^2u_{l}}{dr^2}w_{\lambda l} - u_{l}\frac{d^2w_{\lambda l}}{dr^2} \right) dr \\
& = \int_0^a\left( \left[k^2-\frac{2\mu}{\hbar^2}(V+iW)-\frac{l(l+1)}{r^2}\right]u_lw_{\lambda l} + u_l\left[k_\lambda^2-\frac{2\mu}{\hbar^2}V-\frac{l(l+1)}{r^2}\right]w_{\lambda l} \right)dr \\
& = (k_\lambda^2 - k^2) \int_0^a u_lw_{\lambda l}dr + \frac{2\mu}{\hbar}\int_0^aWu_lw_{\lambda l}dr.
\end{aligned}
\end{equation}
\noindent
Defining $\overline{W}_{\lambda l}$ as
\begin{equation}
\overline{W}_{\lambda l} = \frac{ \int_0^a Wu_lw_{\lambda l}dr }{ \int_0^a u_lw_{\lambda l}dr }
\end{equation}
\noindent
permits rewriting Eq. (II A2.18)\ref{eq:schroding-plugin} in the form
\begin{equation}\label{eq:schroding-plugin-reduced}
\int_0^a \left( \frac{d^2u_{l}}{dr^2}w_{\lambda l} - u_{l}\frac{d^2w_{\lambda l}}{dr^2} \right) dr = \left( k_\lambda^2 - k^2 + i\frac{2\mu}{\hbar^2}\overline{W}_{\lambda l} \right)\int_0^a u_lw_{\lambda l}dr.
\end{equation}
\noindent
Integrating the left-hand side of this equation gives
\begin{equation}\label{eq:schroding-left}
\begin{aligned}
\int_0^a \left( \frac{d^2u_{l}}{dr^2}w_{\lambda l} - u_{l}\frac{d^2w_{\lambda l}}{dr^2} \right) dr & = \left[ \frac{du_{l}}{dr}w_{\lambda l} - u_{l}\frac{dw_{\lambda l}}{dr} \right]_0^a = \left[ \frac{du_{l}}{dr}w_{\lambda l} - u_{l}\frac{dw_{\lambda l}}{dr} \right]_{r=a} \\
& = \left[ \frac{du_{l}}{dr}w_{\lambda l} - u_{l}\frac{B_l}{a}w_{\lambda l} \right]_{r=a} = \left[ a\frac{du_{l}}{dr} - u_{l}B_l \right]_{r=a} \frac{w_{\lambda l(a)}}{a},
\end{aligned}
\end{equation}
\noindent
in which we have again made use of the boundary condition of Eq. (II A2.9)\ref{eq:schrod-bound-conditions}. Integrating the right-hand side of Eq. (II A2.20)\ref{eq:schroding-plugin-reduced} by applying Eq. (II A2.16)\ref{eq:c-lambda-l} gives
\begin{equation}\label{eq:schroding-right}
\left( k_\lambda^2 - k^2 + i\frac{2\mu}{\hbar^2}\overline{W}_{\lambda l} \right)\int_0^a u_lw_{\lambda l}dr = \left( k_\lambda^2 - k^2 + i\frac{2\mu}{\hbar^2}\overline{W}_{\lambda l} \right)c_{\lambda l}.
\end{equation}
\noindent
Equating Eqs. (II A2.21)\ref{eq:schroding-left} and (II A2.22)\ref{eq:schroding-right} therefore gives
\begin{equation}
\begin{aligned}
\left[ a\frac{du_l}{dr}-u_lB_l \right]_{r=a}\frac{w_{\lambda l}}{a} & = \left( k_\lambda^2 - k^2 + i\frac{2\mu }{\hbar^2}\overline{W}_{\lambda l} \right)c_{\lambda l}, \\
\left[ a\frac{du_l}{dr}-u_lB_l \right]_{r=a}\frac{w_{\lambda l}}{a} & = \left( E^\text{(CoM)}_\lambda - E^\text{(CoM)} + i\overline{W}_{\lambda l} \right)\frac{2\mu c_{\lambda l}}{\hbar^2},
\end{aligned}
\end{equation}
\noindent
or
\begin{equation}
c_{\lambda l} = \frac{\hbar^2w_{\lambda l}(a)}{2\mu a\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{W}_{\lambda l}\right)} \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a}.
\end{equation}
\noindent
Inserting this into Eq. (II A2.15)\ref{eq:internal-wave-func} gives
\begin{equation}
u_l(r) = \sum_{\lambda} w_{\lambda l}(r)\frac{\hbar^2w_{\lambda l}(a)}{2\mu a\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{W}_{\lambda l}\right)} \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a},
\end{equation}
\noindent
which when evaluated at $r = a$ , becomes
\begin{equation}
u_l(a) = \sum_{\lambda} \frac{\hbar^2w_{\lambda l}^2(a)}{2\mu a\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{W}_{\lambda l}\right)} \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a},
\end{equation}
\noindent
Rearranging, this becomes
\begin{equation}\label{eq:u-l-eval-at-bound}
\begin{aligned}
u_l(a) & = \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a} \sum_{\lambda} \frac{\hbar^2w_{\lambda l}^2(a)/2\mu a}{\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{W}_{\lambda l}\right)} \\
& = \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a} \sum_{\lambda} \frac{\gamma_{\lambda l}^2}{\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{\Gamma}_{\lambda l}/2\right)},
\end{aligned}
\end{equation}
\noindent
in which the decay amplitude $\gamma_{\lambda l}$ is defined as
\begin{equation}
\gamma_{\lambda l} \equiv \sqrt{\frac{\hbar^2w_{\lambda l}^2(a)}{2\mu a}}
\end{equation}
\noindent
and the absorption width $\Gamma_{\lambda l}$ as
% NOTE: I'm changing from Nancy's formula here with overline{W}
\begin{equation}
\Gamma_{\lambda l} \equiv 2 \overline{W}_{\lambda l} .
\end{equation}
\noindent
If we then define the $R$-function as
\begin{equation}\label{eq:R-func-center-of-mass}
R_l = \sum_\lambda \frac{\gamma_{\lambda l}^2}{\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{\Gamma}_{\lambda l}/2\right)},
\end{equation}
\noindent
then Eq. (II A2.27)\ref{eq:u-l-eval-at-bound} can be written in the form
\begin{equation}\label{eq:u-l-r-matrix}
u_l = \left(a\frac{du_l}{dr}-u_lB_l\right)R_l,
\end{equation}
\noindent
in which everything is evaluated at the matching radius $a$. Note that the form of Eq. \ref{eq:R-func-center-of-mass} (which is in the CoM frame) is the same as if it were in the laboratory frame of reference. This is because of canceling terms in the numerator and denominator of the R-matrix, e.g.
\begin{equation} \label{eq:lab-to-com-params}
\begin{aligned}
E & \equiv E^{\text{(lab)}} = \frac{M}{m+M}E^{\text{(CoM)}}, \\
E_\lambda & \equiv E^\text{(lab)}_{\lambda} = \frac{M}{m+M}E^\text{(CoM)}_{\lambda}, \\
\gamma_{\lambda,l}^2 & \equiv (\gamma^\text{(lab)}_{\lambda,l})^2 = \frac{M}{m+M}(\gamma^\text{(CoM)}_{\lambda,l})^2, \text{and} \\
\Gamma_{\gamma,l} & \equiv \Gamma^\text{(lab)}_{\gamma,l} = \frac{M}{m+M} \Gamma^\text{(CoM)}_{\gamma,l}.
\end{aligned}
\end{equation}
\noindent
Due to this relationship, Eq. \ref{eq:R-func-center-of-mass} can be used for CoM or laboratory frame parameters. Conventional SAMMY parameterization is to use the laboratory frame.
\noindent
\textbf{Scattering matrix in terms of R-matrix (neutrons only)}
Equation (II A2.31)\ref{eq:u-l-r-matrix} can be converted into the usual R-matrix formulae by inserting Eq. (II A2.5)\ref{eq:in-out-wave-funcs},
\begin{equation}
u_l = I_l - U_l O_l,
\end{equation}
\noindent
yielding
\begin{equation}\label{eq:u-l-expanded}
I_l - U_l O_l = \left[a\left(\frac{dI_l}{dr}-U_l\frac{dO_l}{dr}\right)-B_l(I_l-U_lO_l)\right]R_l,
\end{equation}
\noindent
in which everything is again evaluated at the matching radius $a$. Solving Eq. (II A2.33)\ref{eq:u-l-expanded} for $U$ gives
\begin{equation}\label{eq:u-l-i-l-matching}
U_l\left[-O_l+R_l\left(a\frac{dO_l}{dr}-B_lO_l\right)\right] = I_l-R_l\left(a\frac{dI_l}{dr}-B_lI_l\right),
\end{equation}
\noindent
or
\begin{equation}
U_l = \frac{ I_l-R_l\left(a\frac{dI_l}{dr}-B_lI_l\right) }{ \left[-O_l+R_l\left(a\frac{dO_l}{dr}-B_lO_l\right)\right] } = \frac{I_l}{O_l}\frac{ 1-R_l\left(\frac{a}{I_l}\frac{dI_l}{dr}-B_l\right) }{ 1-R_l\left(\frac{a}{O_l}\frac{dO_l}{dr}-B_l\right) }.
\end{equation}
\noindent
We define $L_l$ as
\begin{equation}
L_l \equiv \frac{a}{O_l(a)}\frac{dO_l}{dr}\Bigr|_{r=a} \equiv S_l + iP_l.
\end{equation}
\noindent
For spinless particles, $I_l^* = O_l$, so that
\begin{equation}
\frac{a}{I_l(a)} \frac{dI_l}{dr}\Bigr|_{r=a} = L_l^* = S_l-iP_l
\end{equation}
\noindent
and
\begin{equation}
\frac{I_l}{O_l} = \frac{O_l^*}{O_l} = \frac{|O|e^{-i\phi}}{|O|e^{i\phi}} = e^{-2i\phi}.
\end{equation}
\noindent
Therefore Eq. (II A2.34)\ref{eq:u-l-i-l-matching} becomes
\begin{equation}
U_l = e^{-2i\phi}\frac{1-R_l(L_l^*-B_l)}{1-R_l(L_l-B_l)},
\end{equation}
\noindent
which is the usual form for the scattering matrix in terms of the R-matrix in this simple case.
\noindent
\textbf{Relating the scattering matrix to the cross sections}
The relationship between the scattering matrix $U$ and the cross section $\sigma$ is also described by many authors; see, for example, [AF71]\cite{foderaro_1971}. Here we provide a summary for the simplest case. The wave function for a spinless particle far from the scattering source can be written as
\begin{equation}
\psi(r,\theta) = e^{ikz} + \frac{e^{ikr}}{r}f(\theta),
\end{equation}
\noindent
where $f$ has the form
\begin{equation}\label{f-theta}
f(\theta) = \frac{1}{2ik}\sum_l(2l+1)\left[U_l-1\right]P_l(\cos\theta).
\end{equation}
\noindent
The cross section is then given by
\begin{equation}\label{eq:angle-diff-xs-to-f-theta}
\frac{d\sigma}{d\Omega} = \left|f(\theta)\right|^2.
\end{equation}
For angle-integrated cross sections, the equation found by inserting Eq. (II A2 a.2)\ref{f-theta} into Eq. (II A2 a.3)\ref{eq:angle-diff-xs-to-f-theta} can be integrated to give
\begin{equation}
\begin{aligned}
\sigma =& \int \left[-\frac{1}{2ik}\sum_l(2l+1)\left[U_l^*-1\right]P_l(\cos\theta)\right] \\
& \times\left[\frac{1}{2ik}\sum_{l'}(2l'+1)\left[U_{l'}-1\right]P_{l'}(\cos\theta)\right]d(\cos\theta)d\phi \\
= & \frac{1}{4k^2}\sum_{ll'}(2l+1)(2l'+1)[U_l^*-1][U_{l'}-1]\int_0^{2\pi}d\phi\int_{-1}^1P_l(\cos\theta)P_{l'}(\cos\theta)d(\cos\theta) \\
= & \frac{1}{4k^2}\sum_{ll'}(2l+1)(2l'+1)[U_l^*-1][U_{l'}-1]2\pi\frac{2}{2l+1}\delta_{ll'} \\
= & \frac{\pi}{k^2}\sum_l(2l+1)|U_l-1|^2 .
\end{aligned}
\end{equation}
This is analogous to the ``standard'' scattering theory equation shown in Eq. (II A.1)\ref{eq:sigma-ccprime}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Versions of Multilevel R-matrix Theory} \label{sec:versions-of-r-matrix}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Many representations of multilevel R-matrix theory have been developed over the years.
For a summary of the more common versions, the reader is referred to the works of Fr\"{o}hner
[FF80, FF00]\cite{frohner1980applied,jeff18_frohner}.
Four versions of R-matrix theory are available in SAMMY: the Reich-Moore
approximation (Section II.B.1)\ref{subsec:r-matrix-RM}, the single-level (SLBW) and multilevel Breit-Wigner (MLBW) approximations (Section II.B.3)\ref{subsec:r-matrix-BW}, and a variant on the Reich Moore which mimics the full R-matrix (Section II.B.2)\ref{subsec:r-matrix-full}. An option to include a direct capture component is also provided (Section II.B.4)\ref{subsec:direct-capture}.
The Reich-Moore approximation is the preferred method for most modern evaluations; it
is the default formalism for SAMMY runs. Fr\"{o}hner, in fact, suggests that the Reich-Moore approximation is universally applicable to all cases: ``Experience has shown that with this approximation [Reich Moore] all resonance cross section data can be described in detail, in the windows as well as in the peaks, even the weirdest multilevel interference patterns...It works equally well for light, medium-mass and heavy nuclei, fissile and nonfissile.'' \cite{jeff18_frohner}(pg. 60). For most purposes, Reich Moore is indeed indistinguishable from the exact formulation. Notable exceptions are situations where interference effects exist between capture and other channels. For those cases, small modifications to the SAMMY input will permit the user to mimic the effect of the non-approximated R-matrix; see Section II.B.2\ref{subsec:r-matrix-full} for details. Occasionally it is not possible to properly describe a cross section within the confines of R-matrix theory, because the reaction includes a direct component. SAMMY has provisions for the user to provide a numerical description of this component; see Section II.B.4\ref{subsec:direct-capture} for details.
Also available within SAMMY are both the SLBW and the MLBW formulations (Section II.B.3)\ref{subsec:r-matrix-BW}; these are included for the sake of completeness, for comparison purposes, and because many of the evaluations in the nuclear data files were performed with Breit-Wigner formulae. However, it is strongly recommended that only Reich Moore be used for new evaluations, for several reasons: MLBW is often inadequate; SLBW is almost always inadequate. When it is correct, MLBW gives identical results to Reich Moore. ``Ease of Programming'' is no longer a valid excuse for using MLBW, since the programming has already been accomplished. Similarly, a slow computer is no longer a legitimate excuse, since modern computers can readily handle the more rigorous formulae.
Finally, it should be noted that SAMMY's implementation of MLBW does not correspond to the usual definition of MLBW. Instead, SAMMY uses the ENDF [ENDF-102]\cite{endf8} convention in which only the elastic cross section is truly multilevel, and all other types of cross section are single level.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Reich-Moore Approximation to Multilevel R-Matrix Theory} \label{subsec:r-matrix-RM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The Reich-Moore approximation \cite{reich_moore} is based on the idea that capture channels behave quite differently from particle channels. The particle-pair configuration for a capture channel consists of a gamma ``particle'' plus a nucleus with one more neutron than the target nucleus. For most physical situations, there are a multitude of such capture channels, whose behavior can be treated in an aggregate or average manner. It is assumed that there is no net interference between the aggregate capture channel and other channels, and the level-level interference of gamma channels is negligible, so that terms describing such interference may be eliminated from the R-matrix formulae. The mathematical derivation of this ``eliminated-channel approximation'' is discussed in Section II.A.1\ref{subsec:r-matrix-a-matrix}.
In the eliminated-channel approximation, the R-matrix of Eq. (II A.6)\ref{eq:W-matrix} (for the spin group defined by total spin $J$ and implicit parity $\pi$) has the form
\begin{equation}\label{eq:RM-r-matrix}
R_{cc'}=\left[\sum_\lambda \frac{\gamma_{\lambda c}\gamma_{\lambda c'}}{E_\lambda-E-i\overline{\Gamma}_{\lambda\gamma}/2} + R_{c}^{ext}\delta_{cc'} \right] \delta_{JJ'} ,
\end{equation}
where all levels (resonances) of that spin group are included in the sum. Subscript $\lambda$ designates the particular level; subscripts $c$ and $c'$ designate channels (including particle pairs and all the relevant quantum numbers). The width $\overline{\Gamma}_{\lambda\gamma}$ occurring in the denominator corresponds to the ``eliminated'' non-interfering capture channels of the Reich-Moore approximation; we use the bar to indicate that this width is treated differently from other ``particle'' widths.
\noindent
The ``external R-function'' $R_{c}^{ext}$ of Eq. (II B1.1)\ref{eq:RM-r-matrix} will be discussed at the end of Section II.B.1.d\ref{subsec:r-matrix-RM}.
The channel width $\Gamma_{\lambda c}$ is given in terms of the reduced-width amplitude $\gamma_{\lambda c}$ by
\begin{equation}\label{eq:observed-to-reduced-width}
\Gamma_{\lambda c} = 2\gamma_{\lambda c}^2P_c(E),
\end{equation}
\noindent
where $P_c$ is the penetrability, whose value is a function of the type of particles in the channel, of the orbital angular momentum $l$, and of the energy $E$. The reduced-width amplitude $\gamma_{\lambda c}$ is always independent of energy, but the width $\Gamma_{\lambda c}$ may depend on energy via the penetration factor. For fission and for gamma channels, Eq. (II B1.2)\ref{eq:observed-to-reduced-width} becomes
\begin{equation}
\Gamma_{\lambda c} = 2\gamma_{\lambda c}^2
\end{equation}
\noindent
that is, the penetrability is effectively 1. (Note: In this manual, the reduced-width amplitude for the eliminated-channel capture width will be denoted by a bar above the symbol $\Gamma$.)
% >>> CHANGE: I'm capitalizing gamma
Cross sections may be calculated by using the above expressions for $R$, with $L$ given by Eq. (II A.7)\ref{eq:L-matrix}, to generate $W$, and from there calculating $U$ and, ultimately, $\sigma$. However, while Eq. (II A.6)\ref{eq:W-matrix} for $W$ is correct, an equivalent form that is computationally more stable \cite{larson_1993} is
\begin{equation}\label{eq:W-in-terms-of-X}
W = I + 2iX,
\end{equation}
\noindent
where $X$ is given in matrix notation by
\begin{equation}\label{eq:X-matrix}
X = P^{1/2}L^{-1}\left(L^{-1}-R\right)^{-1}RP^{1/2}.
\end{equation}
\noindent
When the suppressed indices and implied summations are inserted, the expression for $X$ becomes
\begin{equation}
X_{cc'} = P_c^{1/2}L_c^{-1}\sum_{c''}\left[(L^{-1}-R)^{-1}\right]_{cc''}R_{c''c'}P_{c'}^{1/2}\delta_{JJ'}.
\end{equation}
\noindent
The various cross sections are then written in terms of X.
All calculations internally within SAMMY are expressed in terms of so-called ``u-parameters,'' as distinguished from ``p-parameters,'' which are the input quantities. The u-parameters associated with the resonance p-parameters are as follows:
\begin{equation}
u_{E_\lambda} = \Biggl\{
\begin{aligned}
\sqrt{E_\lambda} \qquad &\text{for} \qquad E_\lambda > 0 \\
-\sqrt{-E_\lambda} \qquad &\text{for} \qquad E_\lambda < 0 \qquad
\end{aligned}
\end{equation}
\begin{equation}
u_{\Gamma_{\lambda c}} = \gamma_{\lambda c} = \Biggl\{
\begin{aligned}
\sqrt{\frac{\Gamma_{\lambda c}}{2P_l(|E_\lambda-\Xi_c|)}} \qquad &\text{if}\qquad \Gamma_{\lambda c} > 0 \\
-\sqrt{\frac{|\Gamma_{\lambda c}|}{2P_l(|E_\lambda-\Xi_c|)}} \qquad &\text{if}\qquad \Gamma_{\lambda c} < 0 \qquad\text{in the PARameter file},\qquad
\end{aligned}
\end{equation}
\noindent
in which $\Xi_c$ is the energy threshold for the channel (Section II.C.2)\ref{subsec:kinematic-conventions}. It is important to note that the partial-width parameter $\Gamma_{\lambda c}$ is always a positive quantity, while the reduced-width amplitude $\gamma_{\lambda c}$ can be either positive or negative. Nevertheless, in the original SAMMY input or output PARameter file (and also in the ENDF File 2 formats\ref{endf8}), partial widths may appear with negative signs. The convention is that the sign given in those files is associated with the amplitude $\gamma_{\lambda c}$ rather than with the partial width $\Gamma_{\lambda c}$.
% >>> THIS NEEDS TO BE UPDATED on release of a new manual (but it's still true)
As of Revision 8 of this document and Release sammy-8.0.0 of the code, the reduced-width amplitudes and square root of resonance energy may be used as input to SAMMY; see Table VI B.2\ref{} for details. To use this option, include the command ``\texttt{REDUCED WIDTH AMPLITudes are used for input}'' in card set 2 of the INPut file. An output file SAMMY.RED is created in this format whenever output file SAMMY.PAR is created.
% >>> CHANGE: I'm not sub-dividing sections any further; just putting headers
\noindent
\textbf{Energy-differential cross sections}
The observable cross sections are found in terms of $X$ by first substituting Eqs. (II A.4\ref{eq:scat-matrix}, II A.5\ref{eq:omega}, and II B1.3\ref{eq:W-in-terms-of-X}) into Eq. (II A.1)\ref{eq:sigma-ccprime}, summing over spin groups (i.e., over $J^\pi$ ), and then summing over all channels corresponding to those particle pairs and spin groups. If $X^r$ represents the real part and $X^i$ the imaginary part of $X$, then the angle-integrated (but energy-differential) cross section for the interaction that leads from particle pair $\alpha$ to particle pair $\alpha'$ has the form
\begin{equation}\label{eq:sigma-in-terms-of-X}
\begin{aligned}
\sigma_{\alpha,\alpha'}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c \Biggl[ \bigl(\sin^2\phi_c(1-2X_{cc}^i) & - X_{cc}^r\sin(2\phi_c)\bigr)\delta_{\alpha,\alpha'} \\
& + \sum_{c'}\left({X^i_{cc'}}^2 + {X^r_{cc'}}^2\right) \Biggr].
\end{aligned}
\end{equation}
\noindent
(This formula is accurate only for cases in which one of particles in $\alpha$ is a neutron; however, both particles in $\alpha'$ may be charged.)
In Eq. (II B1 a.1)\ref{eq:sigma-in-terms-of-X} the summations are over those channels $c$ and $c'$ (of the spin group defined by $J^\pi$) for which the particle pairs are, respectively, $\alpha$ and $\alpha'$. More than one ``incident channel'' $c = (\alpha ,l, s, J)$ can contribute to this cross section, for example when both $l = 0$ and $l = 2$ are possible, or when, in the case of incident neutrons and non-zero spin target nuclei, both channel spins are allowed. Similarly, there may be several ``exit channels'' $c' = (\alpha',l', s', J')$, depending on the particular reaction being calculated (e.g., elastic, inelastic, fission).
The total cross section (for non-Coulomb initial states) is the sum of Eq. (II B1 a.1)\ref{eq:sigma-in-terms-of-X} over all possible final-state particle-pairs $\alpha'$, assuming the scattering matrix is unitary (i.e., assuming that the sum over $c'$ of $|U_{cc'}^2=1$). Written in terms of the $X$ matrix, the total cross section has the form
\begin{equation}\label{eq:tot-in-terms-of-X}
\sigma_{total}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c \left[ \left(\sin^2\phi_c + X^i_{cc}\cos(2\phi_c)- X_{cc}^r\sin(2\phi_c)\right)\right],
\end{equation}
\noindentwhere again the sum over $c$ includes only those channels of the $J^\pi$ spin group for which the particle pair is $\alpha$.
The angle integrated elastic cross section is given by
\begin{equation}\label{eq:elastic-in-terms-of-X}
\begin{aligned}
\sigma_{elastic}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c \Biggl[ \sin^2\phi_c(1-2X^i_{cc}) &-X^r_{cc}\sin(2\phi_c) \\
& +\sum_{c'}\left({X^i_{cc'}}^2+{X^r_{cc'}}^2\right) \Biggr].
\end{aligned}
\end{equation}
\noindent
In this case, both $c$ and $c'$ are limited to those channels of the $J^\pi$ spin group for which the particle-pair is $\alpha$; again, there may be more than one such channel for a given spin group.
Similarly, the reaction cross section from particle pair $\alpha$ to particle pair $\alpha'$ (where $\alpha'$ is not equal to $\alpha$) is
\begin{equation}\label{eq:reax-in-terms-of-X}
\sigma_{reaction}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c\sum_{c'}\left[{X^i_{cc'}}^2+{X^r_{cc'}}^2\right].
\end{equation}
Here $c$ is restricted to those channels of the $J^\pi$ spin group from which the particle pair is $\alpha$, and $c'$ to those channels for which the particle-pair is $\alpha'$.
The absorption cross section has the form
\begin{equation}
\sigma_{absorption}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c\left[X^i_{cc} -\sum_{c'} \left({X^i_{cc'}}^2+{X^r_{cc'}}^2\right)\right].
\end{equation}
\noindent
Here both the sum over $c$ and the sum over $c'$ include all incident particle channels (i.e., particle pair $\alpha$ only) for the $J^\pi$ spin group.
The capture cross section for the eliminated radiation channels can be calculated directly as
\begin{equation}\label{eq:sig-cap-in-terms-of-X}
\sigma_{capture}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_{inc\; c}\left[X^i_{cc} -\sum_{all\; c'} \left({X^i_{cc'}}^2+{X^r_{cc'}}^2\right)\right].
\end{equation}
\noindent
or may be found by subtracting the sum of all reaction cross sections from the absorption cross section. In Eq. (II B1 a.6)\ref{eq:sig-cap-in-terms-of-X}, the sum over $c$ includes all incident particle channels for the $J^\pi$ spin group, and the sum over $c'$ includes \textbf{all} particle channels, both incident and exit, for that spin group.
\newpage
\noindent
\textbf{One-level two-channel case}
For a simple one-level, two-channel case for which the shift factor is set to zero, the various cross sections defined directly above can easily be expressed in terms of resonance parameters. Users are reminded that SAMMY is by no means restricted to this simple case and can be used with as many levels and as many channels as are needed to describe the particular physical situation. Nevertheless, it is useful to examine the cross section equations for this simple case: while these equations are a crude over-simplification for most physical situations, there is often physical insight to be gained by examination of these equations.
For this simple case, the X matrix of Eq. (II B1.4)\ref{eq:X-matrix} takes the form
\begin{equation}
\begin{aligned}
X &= \sqrt{P}L^{-1}(L^{-1}-R)^{-1}R\sqrt{P} \\
&= \begin{bmatrix}
\frac{\sqrt{P_1}}{iP_1} & 0 \\
0 & \frac{\sqrt{P_2}}{iP_2}
\end{bmatrix}
\begin{bmatrix}
\frac{1}{iP_1}-\frac{\gamma_1^2}{D} & -\frac{\gamma_1\gamma_2}{D} \\
-\frac{\gamma_1\gamma_2}{D} & \frac{1}{iP_2}-\frac{\gamma_2^2}{D}
\end{bmatrix}^{-1}
\begin{bmatrix}
\frac{\gamma_1^2}{D} & \frac{\gamma_1\gamma_2}{D} \\
\frac{\gamma_1\gamma_2}{D} & \frac{\gamma_2^2}{D}
\end{bmatrix}
\begin{bmatrix}
\sqrt{P_1} & 0 \\
0 & \sqrt{P_2}
\end{bmatrix},
\end{aligned}
\end{equation}
in which the subscript on the penetrabilities denotes the channel number (not the angular momentum), the symbol $D$ has been used for $E_\lambda-E-i\overline{\gamma}^2_{\lambda\gamma}$, and the subscript $\lambda$ has been omitted from the reduced-width amplitudes for simplicity's sake. This equation can be rewritten as
\begin{equation}
\begin{array}{l}
X=\frac{i P_{1} P_{2} D}{i D}\left[\begin{array}{cc}
\frac{1}{\sqrt{P_{1}}} & 0 \\
0 & \frac{1}{\sqrt{P_{2}}}
\end{array}\right]\left[\begin{array}{cc}
P_{2}\left(D-i P_{1} \gamma_{1}^{2}\right) & -i P_{1} P_{2} \gamma_{1} \gamma_{2} \\
-i P_{1} P_{2} \gamma_{1} \gamma_{2} & P_{1}\left(D-i P_{2} \gamma_{2}^{2}\right)
\end{array}\right]^{-1} \\ \\
\qquad\qquad\times\left[\begin{array}{cc}
\gamma_{1}^{2} & \gamma_{1} \gamma_{2} \\
\gamma_{1} \gamma_{2} & \gamma_{2}^{2}
\end{array}\right]
\left[\begin{array}{cc}
\sqrt{P_{1}} & 0 \\
0 & \sqrt{P_{2}}
\end{array}\right] \\ \\
\quad=\frac{P_{1} P_{2}}{P_{1} P_{2}\left(D^{2}-i P_{1} \gamma_{1}^{2} D-i P_{2} \gamma_{2}^{2} D\right)}\left[\begin{array}{cc}
\frac{1}{\sqrt{P_{1}}} & 0 \\
0 & \frac{1}{\sqrt{P_{2}}}
\end{array}\right]\left[\begin{array}{cc}
P_{1}\left(D-i P_{2} \gamma_{2}^{2}\right) & i P_{1} P_{2} \gamma_{1} \gamma_{2} \\
i P_{1} P_{2} \gamma_{1} \gamma_{2} & P_{2}\left(D-i P_{1} \gamma_{1}^{2}\right)
\end{array}\right] \\ \\
\qquad\qquad\times\left[\begin{array}{cc}
\gamma_{1}^{2} & \gamma_{1} \gamma_{2} \\
\gamma_{1} \gamma_{2} & \gamma_{2}^{2}
\end{array}\right]\left[\begin{array}{cc}
\sqrt{P_{1}} & 0 \\
0 & \sqrt{P_{2}}
\end{array}\right],
\end{array}
\end{equation}
\noindent
or,
\begin{equation}
\begin{aligned}
X &=\frac{1}{\left(D^{2}-i P_{1} \gamma_{1}^{2} D-i P_{2} \gamma_{2}^{2} D\right)}\left[\begin{array}{cc}
\frac{1}{\sqrt{P_{1}}} & 0 \\
0 & \frac{1}{\sqrt{P_{2}}}
\end{array}\right] \\
& \qquad \times\left[\begin{array}{cc}
P_{1} \gamma_{1}^{2} D-i P_{1} P_{2} \gamma_{1}^{2} \gamma_{2}^{2}+i P_{1} P_{2} \gamma_{1}^{2} \gamma_{2}^{2} & P_{1} D \gamma_{1} \gamma_{2}-i P_{1} P_{2} \gamma_{1} \gamma_{2}^{3}+i P_{1} P_{2} \gamma_{1} \gamma_{2}^{3} \\
i P_{1} P_{2} \gamma_{1}^{3} \gamma_{2}+P_{2} D \gamma_{1} \gamma_{2}-i P_{1} P_{2} \gamma_{1}^{3} \gamma_{2} & i P_{1} P_{2} \gamma_{1}^{2} \gamma_{2}^{2}+P_{2} \gamma_{2}^{2} D-i P_{1} P_{2} \gamma_{1}^{2} \gamma_{2}^{2}
\end{array}\right] \\
& \qquad \times\left[\begin{array}{cc}
\sqrt{P_{1}} & 0 \\
0 & \sqrt{P_{2}}
\end{array}\right] \\
& = \frac{1}{\left(D^{2}-i P_{1} \gamma_{1}^{2} D-i P_{2} \gamma_{2}^{2} D\right)}\left[\begin{array}{cc}\frac{1}{\sqrt{P_{1}}} & 0 \\ 0 & \frac{1}{\sqrt{P_{2}}}\end{array}\right]\left[\begin{array}{cc}P_{1} \gamma_{1}^{2} D & P_{1} D \gamma_{1} \gamma_{2} \\ P_{2} D \gamma_{1} \gamma_{2} & P_{2} \gamma_{2}^{2} D\end{array}\right] \\
& \qquad \times\left[\begin{array}{cc}\sqrt{P_{1}} & 0 \\ 0 & \sqrt{P_{2}}\end{array}\right] \\
& =\frac{1}{\left(D-i P_{1} \gamma_{1}^{2}-i P_{2} \gamma_{2}^{2}\right)}\left[\begin{array}{cc}
P_{1} \gamma_{1}^{2} & \sqrt{P_{1} P_{2}} \gamma_{1} \gamma_{2} \\
\sqrt{P_{1} P_{2}} \gamma_{1} \gamma_{2} & P_{2} \gamma_{2}^{2}
\end{array}\right] ,
\end{aligned}
\end{equation}
\noindent
or finally,
\begin{equation}
\begin{aligned} X &=\frac{1}{\left(E_{\lambda}-E-i \bar{\gamma}_{\gamma}^{2}-i P_{1} \gamma_{1}^{2}-i P_{2} \gamma_{2}^{2}\right)}\left[\begin{array}{cc}
P_{1} \gamma_{1}^{2} & \sqrt{P_{1} P_{2}} \gamma_{1} \gamma_{2} \\
\sqrt{P_{1} P_{2}} \gamma_{1} \gamma_{2} & P_{2} \gamma_{2}^{2}
\end{array}\right] \\
&=\frac{1}{\left(E_{\lambda}-E-i \Gamma / 2\right)}\left[\begin{array}{cc}
\Gamma_{1} / 2 & \sqrt{\Gamma_{1} \Gamma_{2}} / 2 \\
\sqrt{\Gamma_{1} \Gamma_{2}} / 2 & \Gamma_{2} / 2
\end{array}\right],
\end{aligned}
\end{equation}
\noindent
in which $\Gamma$ is the sum of the partial widths $\Gamma_1+\Gamma_2+\Gamma_\gamma$.
In this form, $X$ can be substituted into the equations for the various cross sections. Assuming the second channel is a reaction channel, Eq. (II B1 a.2)\ref{eq:tot-in-terms-of-X} for the total cross section becomes
\begin{equation}
\begin{aligned}
\sigma_{\text {total }}(E) &=\frac{4 \pi}{k_{\alpha}^{2}} g_{J}\left[\sin ^{2} \phi_{c}+\frac{\Gamma \Gamma_{1}}{4 d} \cos \left(2 \phi_{c}\right)-\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{2 d} \sin \left(2 \phi_{c}\right)\right] \\
&=\frac{2 \pi}{k_{\alpha}^{2}} g_{J}\left[1-\left(1-\frac{\Gamma \Gamma_{1}}{2 d}\right) \cos \left(2 \phi_{c}\right)-\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{d} \sin \left(2 \phi_{c}\right)\right],
\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].
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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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